id	sid	tid	token	lemma	pos
ejde-761	1	1	electronic	electronic	ADJ
ejde-761	1	2	journal	journal	NOUN
ejde-761	1	3	of	of	ADP
ejde-761	1	4	differential	differential	ADJ
ejde-761	1	5	equations	equation	NOUN
ejde-761	1	6	,	,	PUNCT
ejde-761	1	7	vol	vol	NOUN
ejde-761	1	8	.	.	NOUN
ejde-761	1	9	2024	2024	NUM
ejde-761	1	10	(	(	PUNCT
ejde-761	1	11	2024	2024	NUM
ejde-761	1	12	)	)	PUNCT
ejde-761	1	13	,	,	PUNCT
ejde-761	1	14	no	no	INTJ
ejde-761	1	15	.	.	NOUN
ejde-761	1	16	68	68	NUM
ejde-761	1	17	,	,	PUNCT
ejde-761	1	18	pp	pp	ADJ
ejde-761	1	19	.	.	PUNCT
ejde-761	2	1	1–22	1–22	PROPN
ejde-761	2	2	.	.	PUNCT
ejde-761	3	1	issn	issn	PROPN
ejde-761	3	2	:	:	PUNCT
ejde-761	3	3	1072	1072	NUM
ejde-761	3	4	-	-	SYM
ejde-761	3	5	6691	6691	NUM
ejde-761	3	6	.	.	PUNCT
ejde-761	4	1	url	url	PROPN
ejde-761	4	2	:	:	PUNCT
ejde-761	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-761	4	4	,	,	PUNCT
ejde-761	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-761	4	6	doi	doi	PROPN
ejde-761	4	7	:	:	PUNCT
ejde-761	4	8	10.58997	10.58997	NUM
ejde-761	4	9	/	/	SYM
ejde-761	4	10	ejde.2024.68	ejde.2024.68	NOUN
ejde-761	4	11	instability	instability	NOUN
ejde-761	4	12	of	of	ADP
ejde-761	4	13	energy	energy	NOUN
ejde-761	4	14	solutions	solution	NOUN
ejde-761	4	15	,	,	PUNCT
ejde-761	4	16	travelling	travel	VERB
ejde-761	4	17	waves	wave	NOUN
ejde-761	4	18	,	,	PUNCT
ejde-761	4	19	and	and	CCONJ
ejde-761	4	20	scaling	scale	VERB
ejde-761	4	21	invariance	invariance	NOUN
ejde-761	4	22	for	for	ADP
ejde-761	4	23	a	a	DET
ejde-761	4	24	fourth	fourth	ADJ
ejde-761	4	25	-	-	PUNCT
ejde-761	4	26	order	order	NOUN
ejde-761	4	27	p	p	ADJ
ejde-761	4	28	-	-	PUNCT
ejde-761	4	29	laplacian	laplacian	ADJ
ejde-761	4	30	operator	operator	NOUN
ejde-761	4	31	with	with	ADP
ejde-761	4	32	superlinear	superlinear	NOUN
ejde-761	4	33	reaction	reaction	NOUN
ejde-761	4	34	josé	josé	PROPN
ejde-761	4	35	luis	luis	PROPN
ejde-761	4	36	díaz	díaz	PROPN
ejde-761	4	37	palencia	palencia	PROPN
ejde-761	4	38	abstract	abstract	NOUN
ejde-761	4	39	.	.	PUNCT
ejde-761	5	1	this	this	DET
ejde-761	5	2	analysis	analysis	NOUN
ejde-761	5	3	explores	explore	VERB
ejde-761	5	4	the	the	DET
ejde-761	5	5	oscillatory	oscillatory	ADJ
ejde-761	5	6	behavior	behavior	NOUN
ejde-761	5	7	of	of	ADP
ejde-761	5	8	traveling	travel	VERB
ejde-761	5	9	wave	wave	NOUN
ejde-761	5	10	solutions	solution	NOUN
ejde-761	5	11	for	for	ADP
ejde-761	5	12	a	a	DET
ejde-761	5	13	higher	high	ADJ
ejde-761	5	14	-	-	PUNCT
ejde-761	5	15	order	order	NOUN
ejde-761	5	16	p	p	ADJ
ejde-761	5	17	-	-	PUNCT
ejde-761	5	18	laplacian	laplacian	ADJ
ejde-761	5	19	operator	operator	NOUN
ejde-761	5	20	with	with	ADP
ejde-761	5	21	a	a	DET
ejde-761	5	22	superlinear	superlinear	ADJ
ejde-761	5	23	reaction	reaction	NOUN
ejde-761	5	24	term	term	NOUN
ejde-761	5	25	.	.	PUNCT
ejde-761	6	1	the	the	DET
ejde-761	6	2	study	study	NOUN
ejde-761	6	3	employs	employ	VERB
ejde-761	6	4	an	an	DET
ejde-761	6	5	energy	energy	NOUN
ejde-761	6	6	-	-	PUNCT
ejde-761	6	7	based	base	VERB
ejde-761	6	8	approach	approach	NOUN
ejde-761	6	9	,	,	PUNCT
ejde-761	6	10	incorporating	incorporate	VERB
ejde-761	6	11	generalized	generalize	VERB
ejde-761	6	12	sobolev	sobolev	NOUN
ejde-761	6	13	spaces	space	NOUN
ejde-761	6	14	to	to	PART
ejde-761	6	15	examine	examine	VERB
ejde-761	6	16	relevant	relevant	ADJ
ejde-761	6	17	properties	property	NOUN
ejde-761	6	18	of	of	ADP
ejde-761	6	19	the	the	DET
ejde-761	6	20	solutions	solution	NOUN
ejde-761	6	21	,	,	PUNCT
ejde-761	6	22	including	include	VERB
ejde-761	6	23	oscillations	oscillation	NOUN
ejde-761	6	24	,	,	PUNCT
ejde-761	6	25	diffusive	diffusive	ADJ
ejde-761	6	26	mollification	mollification	NOUN
ejde-761	6	27	,	,	PUNCT
ejde-761	6	28	and	and	CCONJ
ejde-761	6	29	compact	compact	ADJ
ejde-761	6	30	support	support	NOUN
ejde-761	6	31	.	.	PUNCT
ejde-761	7	1	based	base	VERB
ejde-761	7	2	on	on	ADP
ejde-761	7	3	this	this	DET
ejde-761	7	4	energy	energy	NOUN
ejde-761	7	5	framework	framework	NOUN
ejde-761	7	6	,	,	PUNCT
ejde-761	7	7	the	the	DET
ejde-761	7	8	regularity	regularity	NOUN
ejde-761	7	9	of	of	ADP
ejde-761	7	10	the	the	DET
ejde-761	7	11	involved	involved	ADJ
ejde-761	7	12	operator	operator	NOUN
ejde-761	7	13	is	be	AUX
ejde-761	7	14	established	establish	VERB
ejde-761	7	15	.	.	PUNCT
ejde-761	8	1	the	the	DET
ejde-761	8	2	problem	problem	NOUN
ejde-761	8	3	is	be	AUX
ejde-761	8	4	then	then	ADV
ejde-761	8	5	reformulated	reformulate	VERB
ejde-761	8	6	using	use	VERB
ejde-761	8	7	a	a	DET
ejde-761	8	8	traveling	travel	VERB
ejde-761	8	9	wave	wave	NOUN
ejde-761	8	10	approach	approach	NOUN
ejde-761	8	11	,	,	PUNCT
ejde-761	8	12	revealing	reveal	VERB
ejde-761	8	13	the	the	DET
ejde-761	8	14	oscillatory	oscillatory	ADJ
ejde-761	8	15	nature	nature	NOUN
ejde-761	8	16	of	of	ADP
ejde-761	8	17	solutions	solution	NOUN
ejde-761	8	18	near	near	ADP
ejde-761	8	19	the	the	DET
ejde-761	8	20	null	null	ADJ
ejde-761	8	21	solution	solution	NOUN
ejde-761	8	22	.	.	PUNCT
ejde-761	9	1	numerical	numerical	ADJ
ejde-761	9	2	simulations	simulation	NOUN
ejde-761	9	3	are	be	AUX
ejde-761	9	4	conducted	conduct	VERB
ejde-761	9	5	for	for	ADP
ejde-761	9	6	each	each	DET
ejde-761	9	7	wave	wave	NOUN
ejde-761	9	8	speed	speed	NOUN
ejde-761	9	9	to	to	PART
ejde-761	9	10	validate	validate	VERB
ejde-761	9	11	the	the	DET
ejde-761	9	12	analytical	analytical	ADJ
ejde-761	9	13	results	result	NOUN
ejde-761	9	14	,	,	PUNCT
ejde-761	9	15	yielding	yield	VERB
ejde-761	9	16	the	the	DET
ejde-761	9	17	corresponding	corresponding	ADJ
ejde-761	9	18	traveling	travel	VERB
ejde-761	9	19	profiles	profile	NOUN
ejde-761	9	20	.	.	PUNCT
ejde-761	10	1	notably	notably	ADV
ejde-761	10	2	,	,	PUNCT
ejde-761	10	3	one	one	NUM
ejde-761	10	4	of	of	ADP
ejde-761	10	5	the	the	DET
ejde-761	10	6	most	most	ADV
ejde-761	10	7	significant	significant	ADJ
ejde-761	10	8	findings	finding	NOUN
ejde-761	10	9	is	be	AUX
ejde-761	10	10	the	the	DET
ejde-761	10	11	attraction	attraction	NOUN
ejde-761	10	12	towards	towards	ADP
ejde-761	10	13	the	the	DET
ejde-761	10	14	null	null	ADJ
ejde-761	10	15	critical	critical	ADJ
ejde-761	10	16	point	point	NOUN
ejde-761	10	17	,	,	PUNCT
ejde-761	10	18	which	which	PRON
ejde-761	10	19	helps	help	VERB
ejde-761	10	20	prevent	prevent	VERB
ejde-761	10	21	blow	blow	NOUN
ejde-761	10	22	-	-	PUNCT
ejde-761	10	23	up	up	ADP
ejde-761	10	24	formation	formation	NOUN
ejde-761	10	25	.	.	PUNCT
ejde-761	11	1	finally	finally	ADV
ejde-761	11	2	,	,	PUNCT
ejde-761	11	3	the	the	DET
ejde-761	11	4	study	study	NOUN
ejde-761	11	5	delves	delve	VERB
ejde-761	11	6	into	into	ADP
ejde-761	11	7	the	the	DET
ejde-761	11	8	equation	equation	NOUN
ejde-761	11	9	’s	’s	PART
ejde-761	11	10	scale	scale	NOUN
ejde-761	11	11	-	-	PUNCT
ejde-761	11	12	invariant	invariant	ADJ
ejde-761	11	13	properties	property	NOUN
ejde-761	11	14	,	,	PUNCT
ejde-761	11	15	leading	lead	VERB
ejde-761	11	16	to	to	ADP
ejde-761	11	17	the	the	DET
ejde-761	11	18	derivation	derivation	NOUN
ejde-761	11	19	of	of	ADP
ejde-761	11	20	self	self	NOUN
ejde-761	11	21	-	-	PUNCT
ejde-761	11	22	similar	similar	ADJ
ejde-761	11	23	solutions	solution	NOUN
ejde-761	11	24	.	.	PUNCT
ejde-761	12	1	1	1	X
ejde-761	12	2	.	.	X
ejde-761	12	3	problem	problem	NOUN
ejde-761	12	4	description	description	NOUN
ejde-761	12	5	and	and	CCONJ
ejde-761	12	6	objectives	objective	VERB
ejde-761	12	7	reaction	reaction	NOUN
ejde-761	12	8	-	-	PUNCT
ejde-761	12	9	diffusion	diffusion	NOUN
ejde-761	12	10	problems	problem	NOUN
ejde-761	12	11	have	have	AUX
ejde-761	12	12	been	be	AUX
ejde-761	12	13	studied	study	VERB
ejde-761	12	14	using	use	VERB
ejde-761	12	15	various	various	ADJ
ejde-761	12	16	forms	form	NOUN
ejde-761	12	17	of	of	ADP
ejde-761	12	18	diffusive	diffusive	ADJ
ejde-761	12	19	operators	operator	NOUN
ejde-761	12	20	,	,	PUNCT
ejde-761	12	21	including	include	VERB
ejde-761	12	22	the	the	DET
ejde-761	12	23	classical	classical	ADJ
ejde-761	12	24	gaussian	gaussian	ADJ
ejde-761	12	25	second	second	ADJ
ejde-761	12	26	-	-	PUNCT
ejde-761	12	27	order	order	NOUN
ejde-761	12	28	operator	operator	NOUN
ejde-761	12	29	,	,	PUNCT
ejde-761	12	30	p	p	NOUN
ejde-761	12	31	-	-	PUNCT
ejde-761	12	32	laplacian	laplacian	ADJ
ejde-761	12	33	,	,	PUNCT
ejde-761	12	34	higher	high	ADJ
ejde-761	12	35	-	-	PUNCT
ejde-761	12	36	order	order	NOUN
ejde-761	12	37	spatial	spatial	ADJ
ejde-761	12	38	derivatives	derivative	NOUN
ejde-761	12	39	,	,	PUNCT
ejde-761	12	40	porous	porous	ADJ
ejde-761	12	41	medium	medium	NOUN
ejde-761	12	42	,	,	PUNCT
ejde-761	12	43	and	and	CCONJ
ejde-761	12	44	thin	thin	ADJ
ejde-761	12	45	film	film	NOUN
ejde-761	12	46	operators	operator	NOUN
ejde-761	12	47	,	,	PUNCT
ejde-761	12	48	which	which	PRON
ejde-761	12	49	are	be	AUX
ejde-761	12	50	among	among	ADP
ejde-761	12	51	the	the	DET
ejde-761	12	52	most	most	ADV
ejde-761	12	53	notable	notable	ADJ
ejde-761	12	54	.	.	PUNCT
ejde-761	13	1	the	the	DET
ejde-761	13	2	choice	choice	NOUN
ejde-761	13	3	of	of	ADP
ejde-761	13	4	a	a	DET
ejde-761	13	5	specific	specific	ADJ
ejde-761	13	6	diffusion	diffusion	NOUN
ejde-761	13	7	model	model	NOUN
ejde-761	13	8	for	for	ADP
ejde-761	13	9	a	a	DET
ejde-761	13	10	given	give	VERB
ejde-761	13	11	problem	problem	NOUN
ejde-761	13	12	necessitates	necessitate	VERB
ejde-761	13	13	a	a	DET
ejde-761	13	14	deep	deep	ADJ
ejde-761	13	15	understanding	understanding	NOUN
ejde-761	13	16	of	of	ADP
ejde-761	13	17	the	the	DET
ejde-761	13	18	underlying	underlying	ADJ
ejde-761	13	19	mechanisms	mechanism	NOUN
ejde-761	13	20	involved	involve	VERB
ejde-761	13	21	in	in	ADP
ejde-761	13	22	the	the	DET
ejde-761	13	23	exchange	exchange	NOUN
ejde-761	13	24	of	of	ADP
ejde-761	13	25	molecules	molecule	NOUN
ejde-761	13	26	,	,	PUNCT
ejde-761	13	27	temperature	temperature	NOUN
ejde-761	13	28	,	,	PUNCT
ejde-761	13	29	or	or	CCONJ
ejde-761	13	30	energy	energy	NOUN
ejde-761	13	31	.	.	PUNCT
ejde-761	14	1	in	in	ADP
ejde-761	14	2	many	many	ADJ
ejde-761	14	3	studies	study	NOUN
ejde-761	14	4	,	,	PUNCT
ejde-761	14	5	diffusion	diffusion	NOUN
ejde-761	14	6	has	have	AUX
ejde-761	14	7	been	be	AUX
ejde-761	14	8	modeled	model	VERB
ejde-761	14	9	using	use	VERB
ejde-761	14	10	the	the	DET
ejde-761	14	11	concept	concept	NOUN
ejde-761	14	12	of	of	ADP
ejde-761	14	13	random	random	ADJ
ejde-761	14	14	walks	walk	NOUN
ejde-761	14	15	of	of	ADP
ejde-761	14	16	particles	particle	NOUN
ejde-761	14	17	(	(	PUNCT
ejde-761	14	18	for	for	ADP
ejde-761	14	19	a	a	DET
ejde-761	14	20	comprehensive	comprehensive	ADJ
ejde-761	14	21	discussion	discussion	NOUN
ejde-761	14	22	,	,	PUNCT
ejde-761	14	23	refer	refer	VERB
ejde-761	14	24	to	to	ADP
ejde-761	14	25	[	[	X
ejde-761	14	26	37	37	NUM
ejde-761	14	27	]	]	PUNCT
ejde-761	14	28	and	and	CCONJ
ejde-761	14	29	the	the	DET
ejde-761	14	30	references	reference	NOUN
ejde-761	14	31	therein	therein	ADV
ejde-761	14	32	)	)	PUNCT
ejde-761	14	33	.	.	PUNCT
ejde-761	15	1	the	the	DET
ejde-761	15	2	random	random	ADJ
ejde-761	15	3	walk	walk	NOUN
ejde-761	15	4	framework	framework	NOUN
ejde-761	15	5	allows	allow	VERB
ejde-761	15	6	for	for	ADP
ejde-761	15	7	a	a	DET
ejde-761	15	8	microscopic	microscopic	ADJ
ejde-761	15	9	-	-	PUNCT
ejde-761	15	10	level	level	NOUN
ejde-761	15	11	description	description	NOUN
ejde-761	15	12	of	of	ADP
ejde-761	15	13	particle	particle	NOUN
ejde-761	15	14	interactions	interaction	NOUN
ejde-761	15	15	and	and	CCONJ
ejde-761	15	16	this	this	DET
ejde-761	15	17	fact	fact	NOUN
ejde-761	15	18	may	may	AUX
ejde-761	15	19	result	result	VERB
ejde-761	15	20	in	in	ADP
ejde-761	15	21	further	further	ADJ
ejde-761	15	22	accurate	accurate	ADJ
ejde-761	15	23	formulations	formulation	NOUN
ejde-761	15	24	of	of	ADP
ejde-761	15	25	diffusion	diffusion	NOUN
ejde-761	15	26	at	at	ADP
ejde-761	15	27	larger	large	ADJ
ejde-761	15	28	scales	scale	NOUN
ejde-761	15	29	.	.	PUNCT
ejde-761	16	1	other	other	ADJ
ejde-761	16	2	significant	significant	ADJ
ejde-761	16	3	approaches	approach	NOUN
ejde-761	16	4	to	to	ADP
ejde-761	16	5	describing	describe	VERB
ejde-761	16	6	diffusion	diffusion	NOUN
ejde-761	16	7	phenomena	phenomenon	NOUN
ejde-761	16	8	can	can	AUX
ejde-761	16	9	be	be	AUX
ejde-761	16	10	found	find	VERB
ejde-761	16	11	in	in	ADP
ejde-761	16	12	[	[	X
ejde-761	16	13	14	14	NUM
ejde-761	16	14	,	,	PUNCT
ejde-761	16	15	15	15	NUM
ejde-761	16	16	]	]	PUNCT
ejde-761	16	17	,	,	PUNCT
ejde-761	16	18	where	where	SCONJ
ejde-761	16	19	the	the	DET
ejde-761	16	20	concept	concept	NOUN
ejde-761	16	21	of	of	ADP
ejde-761	16	22	free	free	ADJ
ejde-761	16	23	energy	energy	NOUN
ejde-761	16	24	,	,	PUNCT
ejde-761	16	25	first	first	ADV
ejde-761	16	26	introduced	introduce	VERB
ejde-761	16	27	by	by	ADP
ejde-761	16	28	landau	landau	NOUN
ejde-761	16	29	and	and	CCONJ
ejde-761	16	30	ginzburg	ginzburg	NOUN
ejde-761	16	31	,	,	PUNCT
ejde-761	16	32	is	be	AUX
ejde-761	16	33	employed	employ	VERB
ejde-761	16	34	.	.	PUNCT
ejde-761	17	1	this	this	DET
ejde-761	17	2	formulation	formulation	NOUN
ejde-761	17	3	generalizes	generalize	VERB
ejde-761	17	4	the	the	DET
ejde-761	17	5	classical	classical	ADJ
ejde-761	17	6	second	second	ADJ
ejde-761	17	7	-	-	PUNCT
ejde-761	17	8	order	order	NOUN
ejde-761	17	9	diffusion	diffusion	NOUN
ejde-761	17	10	by	by	ADP
ejde-761	17	11	postulating	postulate	VERB
ejde-761	17	12	a	a	DET
ejde-761	17	13	motion	motion	NOUN
ejde-761	17	14	energy	energy	NOUN
ejde-761	17	15	that	that	PRON
ejde-761	17	16	results	result	VERB
ejde-761	17	17	in	in	ADP
ejde-761	17	18	a	a	DET
ejde-761	17	19	higher	high	ADJ
ejde-761	17	20	-	-	PUNCT
ejde-761	17	21	order	order	NOUN
ejde-761	17	22	operator	operator	NOUN
ejde-761	17	23	.	.	PUNCT
ejde-761	18	1	for	for	ADP
ejde-761	18	2	example	example	NOUN
ejde-761	18	3	,	,	PUNCT
ejde-761	18	4	the	the	DET
ejde-761	18	5	analysis	analysis	NOUN
ejde-761	18	6	in	in	ADP
ejde-761	18	7	[	[	X
ejde-761	18	8	14	14	NUM
ejde-761	18	9	]	]	PUNCT
ejde-761	18	10	provided	provide	VERB
ejde-761	18	11	a	a	DET
ejde-761	18	12	non	non	ADJ
ejde-761	18	13	-	-	ADJ
ejde-761	18	14	homogeneous	homogeneous	ADJ
ejde-761	18	15	diffusion	diffusion	NOUN
ejde-761	18	16	expression	expression	NOUN
ejde-761	18	17	derived	derive	VERB
ejde-761	18	18	from	from	ADP
ejde-761	18	19	the	the	DET
ejde-761	18	20	free	free	ADJ
ejde-761	18	21	energy	energy	NOUN
ejde-761	18	22	concept	concept	NOUN
ejde-761	18	23	.	.	PUNCT
ejde-761	19	1	in	in	ADP
ejde-761	19	2	this	this	DET
ejde-761	19	3	case	case	NOUN
ejde-761	19	4	,	,	PUNCT
ejde-761	19	5	the	the	DET
ejde-761	19	6	authors	author	NOUN
ejde-761	19	7	proposed	propose	VERB
ejde-761	19	8	a	a	DET
ejde-761	19	9	free	free	ADJ
ejde-761	19	10	energy	energy	NOUN
ejde-761	19	11	function	function	NOUN
ejde-761	19	12	2020	2020	NUM
ejde-761	19	13	mathematics	mathematic	NOUN
ejde-761	19	14	subject	subject	ADJ
ejde-761	19	15	classification	classification	NOUN
ejde-761	19	16	.	.	PUNCT
ejde-761	20	1	35k92	35k92	NUM
ejde-761	20	2	,	,	PUNCT
ejde-761	20	3	35k91	35k91	NUM
ejde-761	20	4	,	,	PUNCT
ejde-761	20	5	35k55	35k55	NUM
ejde-761	20	6	.	.	PUNCT
ejde-761	21	1	key	key	ADJ
ejde-761	21	2	words	word	NOUN
ejde-761	21	3	and	and	CCONJ
ejde-761	21	4	phrases	phrase	NOUN
ejde-761	21	5	.	.	PUNCT
ejde-761	22	1	higher	high	ADJ
ejde-761	22	2	order	order	NOUN
ejde-761	22	3	p	p	X
ejde-761	22	4	-	-	PUNCT
ejde-761	22	5	laplacian	laplacian	ADJ
ejde-761	22	6	operator	operator	NOUN
ejde-761	22	7	;	;	PUNCT
ejde-761	22	8	travelling	travel	VERB
ejde-761	22	9	waves	wave	NOUN
ejde-761	22	10	;	;	PUNCT
ejde-761	22	11	homotopy	homotopy	NOUN
ejde-761	22	12	;	;	PUNCT
ejde-761	22	13	superlinear	superlinear	NOUN
ejde-761	22	14	reaction	reaction	NOUN
ejde-761	22	15	.	.	PUNCT
ejde-761	23	1	©	©	PROPN
ejde-761	23	2	2024	2024	NUM
ejde-761	23	3	.	.	PUNCT
ejde-761	24	1	this	this	DET
ejde-761	24	2	work	work	NOUN
ejde-761	24	3	is	be	AUX
ejde-761	24	4	licensed	license	VERB
ejde-761	24	5	under	under	ADP
ejde-761	24	6	a	a	DET
ejde-761	24	7	cc	cc	NOUN
ejde-761	24	8	by	by	ADP
ejde-761	24	9	4.0	4.0	NUM
ejde-761	24	10	license	license	NOUN
ejde-761	24	11	.	.	PUNCT
ejde-761	25	1	submitted	submit	VERB
ejde-761	25	2	june	june	PROPN
ejde-761	25	3	18	18	NUM
ejde-761	25	4	,	,	PUNCT
ejde-761	25	5	2024	2024	NUM
ejde-761	25	6	.	.	PUNCT
ejde-761	26	1	published	publish	VERB
ejde-761	26	2	november	november	PROPN
ejde-761	26	3	7	7	NUM
ejde-761	26	4	,	,	PUNCT
ejde-761	26	5	2024	2024	NUM
ejde-761	26	6	.	.	PUNCT
ejde-761	26	7	1	1	NUM
ejde-761	26	8	2	2	NUM
ejde-761	26	9	j.	j.	PROPN
ejde-761	26	10	l.	l.	PROPN
ejde-761	26	11	díaz	díaz	PROPN
ejde-761	26	12	palencia	palencia	PROPN
ejde-761	26	13	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	26	14	dependent	dependent	ADJ
ejde-761	26	15	on	on	ADP
ejde-761	26	16	the	the	DET
ejde-761	26	17	gradient	gradient	NOUN
ejde-761	26	18	of	of	ADP
ejde-761	26	19	the	the	DET
ejde-761	26	20	substance	substance	NOUN
ejde-761	26	21	concentration	concentration	NOUN
ejde-761	26	22	1	1	NUM
ejde-761	26	23	2k(∇v)2	2k(∇v)2	NUM
ejde-761	26	24	,	,	PUNCT
ejde-761	26	25	being	be	AUX
ejde-761	26	26	v	v	ADP
ejde-761	26	27	the	the	DET
ejde-761	26	28	particles	particle	NOUN
ejde-761	26	29	concentration	concentration	NOUN
ejde-761	26	30	.	.	PUNCT
ejde-761	27	1	by	by	ADP
ejde-761	27	2	transforming	transform	VERB
ejde-761	27	3	the	the	DET
ejde-761	27	4	energy	energy	NOUN
ejde-761	27	5	into	into	ADP
ejde-761	27	6	motion	motion	NOUN
ejde-761	27	7	gradients	gradient	NOUN
ejde-761	27	8	through	through	ADP
ejde-761	27	9	the	the	DET
ejde-761	27	10	use	use	NOUN
ejde-761	27	11	of	of	ADP
ejde-761	27	12	chemical	chemical	ADJ
ejde-761	27	13	potential	potential	NOUN
ejde-761	27	14	,	,	PUNCT
ejde-761	27	15	the	the	DET
ejde-761	27	16	authors	author	NOUN
ejde-761	27	17	arrived	arrive	VERB
ejde-761	27	18	at	at	ADP
ejde-761	27	19	a	a	DET
ejde-761	27	20	fourth	fourth	ADJ
ejde-761	27	21	-	-	PUNCT
ejde-761	27	22	order	order	NOUN
ejde-761	27	23	operator	operator	NOUN
ejde-761	27	24	.	.	PUNCT
ejde-761	28	1	the	the	DET
ejde-761	28	2	properties	property	NOUN
ejde-761	28	3	of	of	ADP
ejde-761	28	4	this	this	DET
ejde-761	28	5	non	non	ADJ
ejde-761	28	6	-	-	ADJ
ejde-761	28	7	homogeneous	homogeneous	ADJ
ejde-761	28	8	operator	operator	NOUN
ejde-761	28	9	suggest	suggest	VERB
ejde-761	28	10	a	a	DET
ejde-761	28	11	loss	loss	NOUN
ejde-761	28	12	of	of	ADP
ejde-761	28	13	regularity	regularity	NOUN
ejde-761	28	14	,	,	PUNCT
ejde-761	28	15	particularly	particularly	ADV
ejde-761	28	16	posing	pose	VERB
ejde-761	28	17	challenges	challenge	NOUN
ejde-761	28	18	in	in	ADP
ejde-761	28	19	establishing	establish	VERB
ejde-761	28	20	a	a	DET
ejde-761	28	21	maximum	maximum	ADJ
ejde-761	28	22	principle	principle	NOUN
ejde-761	28	23	.	.	PUNCT
ejde-761	29	1	these	these	DET
ejde-761	29	2	aspects	aspect	NOUN
ejde-761	29	3	of	of	ADP
ejde-761	29	4	regularity	regularity	NOUN
ejde-761	29	5	loss	loss	NOUN
ejde-761	29	6	have	have	AUX
ejde-761	29	7	been	be	AUX
ejde-761	29	8	thoroughly	thoroughly	ADV
ejde-761	29	9	discussed	discuss	VERB
ejde-761	29	10	in	in	ADP
ejde-761	29	11	several	several	ADJ
ejde-761	29	12	studies	study	NOUN
ejde-761	29	13	,	,	PUNCT
ejde-761	29	14	as	as	SCONJ
ejde-761	29	15	referenced	reference	VERB
ejde-761	29	16	in	in	ADP
ejde-761	29	17	[	[	X
ejde-761	29	18	19	19	NUM
ejde-761	29	19	,	,	PUNCT
ejde-761	29	20	25	25	NUM
ejde-761	29	21	,	,	PUNCT
ejde-761	29	22	38	38	NUM
ejde-761	29	23	]	]	PUNCT
ejde-761	29	24	.	.	PUNCT
ejde-761	30	1	there	there	PRON
ejde-761	30	2	are	be	VERB
ejde-761	30	3	other	other	ADJ
ejde-761	30	4	noteworthy	noteworthy	ADJ
ejde-761	30	5	applications	application	NOUN
ejde-761	30	6	where	where	SCONJ
ejde-761	30	7	non	non	ADJ
ejde-761	30	8	-	-	ADJ
ejde-761	30	9	regular	regular	ADJ
ejde-761	30	10	diffusion	diffusion	NOUN
ejde-761	30	11	principles	principle	NOUN
ejde-761	30	12	have	have	AUX
ejde-761	30	13	been	be	AUX
ejde-761	30	14	considered	consider	VERB
ejde-761	30	15	.	.	PUNCT
ejde-761	31	1	for	for	ADP
ejde-761	31	2	example	example	NOUN
ejde-761	31	3	,	,	PUNCT
ejde-761	31	4	the	the	DET
ejde-761	31	5	keller	keller	PROPN
ejde-761	31	6	-	-	PUNCT
ejde-761	31	7	segel	segel	PROPN
ejde-761	31	8	equation	equation	NOUN
ejde-761	31	9	,	,	PUNCT
ejde-761	31	10	which	which	PRON
ejde-761	31	11	is	be	AUX
ejde-761	31	12	significant	significant	ADJ
ejde-761	31	13	in	in	ADP
ejde-761	31	14	biology	biology	NOUN
ejde-761	31	15	for	for	ADP
ejde-761	31	16	modeling	model	VERB
ejde-761	31	17	cell	cell	NOUN
ejde-761	31	18	motion	motion	NOUN
ejde-761	31	19	through	through	ADP
ejde-761	31	20	chemotaxis	chemotaxis	ADJ
ejde-761	31	21	,	,	PUNCT
ejde-761	31	22	is	be	AUX
ejde-761	31	23	one	one	NUM
ejde-761	31	24	such	such	ADJ
ejde-761	31	25	case	case	NOUN
ejde-761	31	26	[	[	X
ejde-761	31	27	30	30	NUM
ejde-761	31	28	]	]	PUNCT
ejde-761	31	29	.	.	PUNCT
ejde-761	32	1	furthermore	furthermore	ADV
ejde-761	32	2	,	,	PUNCT
ejde-761	32	3	various	various	ADJ
ejde-761	32	4	regularity	regularity	NOUN
ejde-761	32	5	analyses	analysis	NOUN
ejde-761	32	6	have	have	AUX
ejde-761	32	7	been	be	AUX
ejde-761	32	8	conducted	conduct	VERB
ejde-761	32	9	to	to	PART
ejde-761	32	10	obtain	obtain	VERB
ejde-761	32	11	smooth	smooth	ADJ
ejde-761	32	12	solutions	solution	NOUN
ejde-761	32	13	to	to	ADP
ejde-761	32	14	the	the	DET
ejde-761	32	15	complex	complex	ADJ
ejde-761	32	16	dynamics	dynamic	NOUN
ejde-761	32	17	of	of	ADP
ejde-761	32	18	chemotaxis	chemotaxis	NOUN
ejde-761	32	19	,	,	PUNCT
ejde-761	32	20	which	which	PRON
ejde-761	32	21	involve	involve	VERB
ejde-761	32	22	diffusion	diffusion	NOUN
ejde-761	32	23	,	,	PUNCT
ejde-761	32	24	reaction	reaction	NOUN
ejde-761	32	25	,	,	PUNCT
ejde-761	32	26	and	and	CCONJ
ejde-761	32	27	absorption	absorption	NOUN
ejde-761	32	28	processes	process	NOUN
ejde-761	32	29	(	(	PUNCT
ejde-761	32	30	see	see	VERB
ejde-761	32	31	[	[	X
ejde-761	32	32	2	2	NUM
ejde-761	32	33	,	,	PUNCT
ejde-761	32	34	13	13	NUM
ejde-761	32	35	,	,	PUNCT
ejde-761	32	36	44	44	NUM
ejde-761	32	37	,	,	PUNCT
ejde-761	32	38	45	45	NUM
ejde-761	32	39	]	]	PUNCT
ejde-761	32	40	)	)	PUNCT
ejde-761	32	41	.	.	PUNCT
ejde-761	33	1	in	in	ADP
ejde-761	33	2	other	other	ADJ
ejde-761	33	3	fields	field	NOUN
ejde-761	33	4	,	,	PUNCT
ejde-761	33	5	non	non	ADJ
ejde-761	33	6	-	-	ADJ
ejde-761	33	7	homogeneous	homogeneous	ADJ
ejde-761	33	8	diffusion	diffusion	NOUN
ejde-761	33	9	,	,	PUNCT
ejde-761	33	10	such	such	ADJ
ejde-761	33	11	as	as	ADP
ejde-761	33	12	that	that	PRON
ejde-761	33	13	of	of	ADP
ejde-761	33	14	the	the	DET
ejde-761	33	15	porous	porous	ADJ
ejde-761	33	16	medium	medium	NOUN
ejde-761	33	17	type	type	NOUN
ejde-761	33	18	,	,	PUNCT
ejde-761	33	19	has	have	AUX
ejde-761	33	20	been	be	AUX
ejde-761	33	21	used	use	VERB
ejde-761	33	22	to	to	PART
ejde-761	33	23	model	model	VERB
ejde-761	33	24	coagulation	coagulation	NOUN
ejde-761	33	25	phenomena	phenomenon	NOUN
ejde-761	33	26	in	in	ADP
ejde-761	33	27	complex	complex	ADJ
ejde-761	33	28	vessel	vessel	NOUN
ejde-761	33	29	geometries	geometry	NOUN
ejde-761	33	30	[	[	X
ejde-761	33	31	9	9	NUM
ejde-761	33	32	]	]	PUNCT
ejde-761	33	33	and	and	CCONJ
ejde-761	33	34	to	to	PART
ejde-761	33	35	simulate	simulate	VERB
ejde-761	33	36	porosity	porosity	NOUN
ejde-761	33	37	in	in	ADP
ejde-761	33	38	peristaltic	peristaltic	ADJ
ejde-761	33	39	transportation	transportation	NOUN
ejde-761	33	40	within	within	ADP
ejde-761	33	41	jeffrey	jeffrey	NOUN
ejde-761	33	42	-	-	PUNCT
ejde-761	33	43	type	type	NOUN
ejde-761	33	44	fluids	fluid	NOUN
ejde-761	33	45	[	[	X
ejde-761	33	46	20	20	NUM
ejde-761	33	47	]	]	PUNCT
ejde-761	33	48	.	.	PUNCT
ejde-761	34	1	these	these	DET
ejde-761	34	2	studies	study	NOUN
ejde-761	34	3	underscore	underscore	VERB
ejde-761	34	4	the	the	DET
ejde-761	34	5	importance	importance	NOUN
ejde-761	34	6	of	of	ADP
ejde-761	34	7	examining	examine	VERB
ejde-761	34	8	the	the	DET
ejde-761	34	9	specific	specific	ADJ
ejde-761	34	10	diffusion	diffusion	NOUN
ejde-761	34	11	characteristics	characteristic	NOUN
ejde-761	34	12	of	of	ADP
ejde-761	34	13	a	a	DET
ejde-761	34	14	phenomenon	phenomenon	NOUN
ejde-761	34	15	when	when	SCONJ
ejde-761	34	16	developing	develop	VERB
ejde-761	34	17	a	a	DET
ejde-761	34	18	model	model	NOUN
ejde-761	34	19	.	.	PUNCT
ejde-761	35	1	it	it	PRON
ejde-761	35	2	is	be	AUX
ejde-761	35	3	worth	worth	ADJ
ejde-761	35	4	noting	note	VERB
ejde-761	35	5	that	that	SCONJ
ejde-761	35	6	diffusion	diffusion	NOUN
ejde-761	35	7	is	be	AUX
ejde-761	35	8	typically	typically	ADV
ejde-761	35	9	formulated	formulate	VERB
ejde-761	35	10	using	use	VERB
ejde-761	35	11	the	the	DET
ejde-761	35	12	regular	regular	ADJ
ejde-761	35	13	gaussian	gaussian	ADJ
ejde-761	35	14	operator	operator	NOUN
ejde-761	35	15	derived	derive	VERB
ejde-761	35	16	from	from	ADP
ejde-761	35	17	the	the	DET
ejde-761	35	18	classical	classical	ADJ
ejde-761	35	19	fick	fick	NOUN
ejde-761	35	20	’s	’s	PART
ejde-761	35	21	law	law	NOUN
ejde-761	35	22	.	.	PUNCT
ejde-761	36	1	however	however	ADV
ejde-761	36	2	,	,	PUNCT
ejde-761	36	3	the	the	DET
ejde-761	36	4	aforementioned	aforementioned	ADJ
ejde-761	36	5	studies	study	NOUN
ejde-761	36	6	highlight	highlight	VERB
ejde-761	36	7	the	the	DET
ejde-761	36	8	value	value	NOUN
ejde-761	36	9	of	of	ADP
ejde-761	36	10	exploring	explore	VERB
ejde-761	36	11	alternative	alternative	ADJ
ejde-761	36	12	diffusion	diffusion	NOUN
ejde-761	36	13	mechanisms	mechanism	NOUN
ejde-761	36	14	and	and	CCONJ
ejde-761	36	15	gaining	gain	VERB
ejde-761	36	16	a	a	DET
ejde-761	36	17	deeper	deep	ADJ
ejde-761	36	18	understanding	understanding	NOUN
ejde-761	36	19	of	of	ADP
ejde-761	36	20	their	their	PRON
ejde-761	36	21	mathematical	mathematical	ADJ
ejde-761	36	22	properties	property	NOUN
ejde-761	36	23	.	.	PUNCT
ejde-761	37	1	the	the	DET
ejde-761	37	2	p	p	PROPN
ejde-761	37	3	-	-	PUNCT
ejde-761	37	4	laplacian	laplacian	ADJ
ejde-761	37	5	operator	operator	NOUN
ejde-761	37	6	has	have	AUX
ejde-761	37	7	been	be	AUX
ejde-761	37	8	widely	widely	ADV
ejde-761	37	9	applied	apply	VERB
ejde-761	37	10	in	in	ADP
ejde-761	37	11	physics	physics	NOUN
ejde-761	37	12	,	,	PUNCT
ejde-761	37	13	chemistry	chemistry	NOUN
ejde-761	37	14	,	,	PUNCT
ejde-761	37	15	and	and	CCONJ
ejde-761	37	16	engineering	engineering	NOUN
ejde-761	37	17	.	.	PUNCT
ejde-761	38	1	as	as	ADP
ejde-761	38	2	a	a	DET
ejde-761	38	3	representative	representative	ADJ
ejde-761	38	4	example	example	NOUN
ejde-761	38	5	,	,	PUNCT
ejde-761	38	6	in	in	ADP
ejde-761	38	7	[	[	PUNCT
ejde-761	38	8	10	10	NUM
ejde-761	38	9	]	]	PUNCT
ejde-761	38	10	,	,	PUNCT
ejde-761	38	11	the	the	DET
ejde-761	38	12	authors	author	NOUN
ejde-761	38	13	provide	provide	VERB
ejde-761	38	14	numerical	numerical	ADJ
ejde-761	38	15	and	and	CCONJ
ejde-761	38	16	analytical	analytical	ADJ
ejde-761	38	17	findings	finding	NOUN
ejde-761	38	18	to	to	PART
ejde-761	38	19	model	model	VERB
ejde-761	38	20	in	in	ADP
ejde-761	38	21	fluid	fluid	ADJ
ejde-761	38	22	mechanics	mechanic	NOUN
ejde-761	38	23	with	with	ADP
ejde-761	38	24	a	a	DET
ejde-761	38	25	p	p	ADJ
ejde-761	38	26	-	-	PUNCT
ejde-761	38	27	laplacian	laplacian	ADJ
ejde-761	38	28	operator	operator	NOUN
ejde-761	38	29	.	.	PUNCT
ejde-761	39	1	it	it	PRON
ejde-761	39	2	is	be	AUX
ejde-761	39	3	of	of	ADP
ejde-761	39	4	interest	interest	NOUN
ejde-761	39	5	to	to	PART
ejde-761	39	6	mention	mention	VERB
ejde-761	39	7	the	the	DET
ejde-761	39	8	p	p	ADJ
ejde-761	39	9	-	-	PUNCT
ejde-761	39	10	laplacian	laplacian	ADJ
ejde-761	39	11	formulation	formulation	NOUN
ejde-761	39	12	in	in	ADP
ejde-761	39	13	emden	emden	ADJ
ejde-761	39	14	-	-	PUNCT
ejde-761	39	15	fowler	fowler	NOUN
ejde-761	39	16	equation	equation	NOUN
ejde-761	39	17	(	(	PUNCT
ejde-761	39	18	see	see	VERB
ejde-761	39	19	[	[	X
ejde-761	39	20	12	12	NUM
ejde-761	39	21	]	]	PUNCT
ejde-761	39	22	)	)	PUNCT
ejde-761	39	23	and	and	CCONJ
ejde-761	39	24	the	the	DET
ejde-761	39	25	study	study	NOUN
ejde-761	39	26	of	of	ADP
ejde-761	39	27	p	p	NOUN
ejde-761	39	28	-	-	PUNCT
ejde-761	39	29	laplacian	laplacian	NOUN
ejde-761	39	30	with	with	ADP
ejde-761	39	31	heterogeneous	heterogeneous	ADJ
ejde-761	39	32	reaction	reaction	NOUN
ejde-761	39	33	(	(	PUNCT
ejde-761	39	34	see	see	VERB
ejde-761	39	35	[	[	X
ejde-761	39	36	17	17	NUM
ejde-761	39	37	]	]	NUM
ejde-761	39	38	)	)	PUNCT
ejde-761	39	39	.	.	PUNCT
ejde-761	40	1	in	in	ADP
ejde-761	40	2	the	the	DET
ejde-761	40	3	presented	present	VERB
ejde-761	40	4	analysis	analysis	NOUN
ejde-761	40	5	,	,	PUNCT
ejde-761	40	6	a	a	DET
ejde-761	40	7	p	p	ADJ
ejde-761	40	8	-	-	PUNCT
ejde-761	40	9	laplacian	laplacian	ADJ
ejde-761	40	10	diffusion	diffusion	NOUN
ejde-761	40	11	of	of	ADP
ejde-761	40	12	higher	high	ADJ
ejde-761	40	13	order	order	NOUN
ejde-761	40	14	is	be	AUX
ejde-761	40	15	considered	consider	VERB
ejde-761	40	16	.	.	PUNCT
ejde-761	41	1	the	the	DET
ejde-761	41	2	single	single	ADJ
ejde-761	41	3	p	p	ADJ
ejde-761	41	4	-	-	PUNCT
ejde-761	41	5	laplacian	laplacian	ADJ
ejde-761	41	6	operator	operator	NOUN
ejde-761	41	7	exhibits	exhibit	VERB
ejde-761	41	8	the	the	DET
ejde-761	41	9	property	property	NOUN
ejde-761	41	10	of	of	ADP
ejde-761	41	11	finite	finite	ADJ
ejde-761	41	12	propagation	propagation	NOUN
ejde-761	41	13	in	in	ADP
ejde-761	41	14	compact	compact	ADJ
ejde-761	41	15	supports	support	NOUN
ejde-761	41	16	.	.	PUNCT
ejde-761	42	1	in	in	ADP
ejde-761	42	2	addition	addition	NOUN
ejde-761	42	3	,	,	PUNCT
ejde-761	42	4	it	it	PRON
ejde-761	42	5	is	be	AUX
ejde-761	42	6	a	a	DET
ejde-761	42	7	monotone	monotone	ADJ
ejde-761	42	8	operator	operator	NOUN
ejde-761	42	9	(	(	PUNCT
ejde-761	42	10	see	see	VERB
ejde-761	42	11	[	[	X
ejde-761	42	12	29	29	NUM
ejde-761	42	13	]	]	PUNCT
ejde-761	42	14	and	and	CCONJ
ejde-761	42	15	references	reference	NOUN
ejde-761	42	16	therein	therein	ADV
ejde-761	42	17	)	)	PUNCT
ejde-761	42	18	.	.	PUNCT
ejde-761	43	1	the	the	DET
ejde-761	43	2	formulated	formulate	VERB
ejde-761	43	3	problem	problem	NOUN
ejde-761	43	4	provides	provide	VERB
ejde-761	43	5	a	a	DET
ejde-761	43	6	superlinear	superlinear	ADJ
ejde-761	43	7	reaction	reaction	NOUN
ejde-761	43	8	for	for	ADP
ejde-761	43	9	which	which	PRON
ejde-761	43	10	blow	blow	NOUN
ejde-761	43	11	-	-	PUNCT
ejde-761	43	12	up	up	ADP
ejde-761	43	13	patterns	pattern	NOUN
ejde-761	43	14	have	have	AUX
ejde-761	43	15	been	be	AUX
ejde-761	43	16	shown	show	VERB
ejde-761	43	17	to	to	PART
ejde-761	43	18	exist	exist	VERB
ejde-761	43	19	(	(	PUNCT
ejde-761	43	20	[	[	X
ejde-761	43	21	23	23	NUM
ejde-761	43	22	]	]	PUNCT
ejde-761	43	23	)	)	PUNCT
ejde-761	43	24	and	and	CCONJ
ejde-761	43	25	is	be	AUX
ejde-761	43	26	given	give	VERB
ejde-761	43	27	by	by	ADP
ejde-761	43	28	:	:	PUNCT
ejde-761	43	29	ut	ut	PROPN
ejde-761	43	30	=	=	SYM
ejde-761	43	31	−∆(|∆u|m∆u	−∆(|∆u|m∆u	X
ejde-761	43	32	)	)	PUNCT
ejde-761	44	1	+	+	CCONJ
ejde-761	44	2	|u|p−1	|u|p−1	NUM
ejde-761	44	3	u	u	NOUN
ejde-761	44	4	,	,	PUNCT
ejde-761	44	5	u0(x	u0(x	NOUN
ejde-761	44	6	)	)	PUNCT
ejde-761	44	7	∈	∈	NOUN
ejde-761	45	1	hn	hn	INTJ
ejde-761	45	2	0	0	NUM
ejde-761	45	3	(	(	PUNCT
ejde-761	45	4	rn	rn	PROPN
ejde-761	45	5	)	)	PUNCT
ejde-761	45	6	,	,	PUNCT
ejde-761	45	7	n	n	X
ejde-761	45	8	≥	≥	NOUN
ejde-761	45	9	1	1	NUM
ejde-761	45	10	,	,	PUNCT
ejde-761	45	11	n	n	CCONJ
ejde-761	45	12	>	>	X
ejde-761	45	13	1	1	NUM
ejde-761	45	14	.	.	PUNCT
ejde-761	45	15	(	(	PUNCT
ejde-761	45	16	1.1	1.1	NUM
ejde-761	45	17	)	)	PUNCT
ejde-761	45	18	note	note	VERB
ejde-761	45	19	that	that	SCONJ
ejde-761	45	20	the	the	DET
ejde-761	45	21	defined	define	VERB
ejde-761	45	22	operator	operator	NOUN
ejde-761	45	23	is	be	AUX
ejde-761	45	24	referred	refer	VERB
ejde-761	45	25	as	as	ADP
ejde-761	45	26	the	the	DET
ejde-761	45	27	fourth	fourth	ADJ
ejde-761	45	28	order	order	NOUN
ejde-761	45	29	p	p	NOUN
ejde-761	45	30	-	-	PUNCT
ejde-761	45	31	laplacian	laplacian	NOUN
ejde-761	45	32	(	(	PUNCT
ejde-761	45	33	see	see	VERB
ejde-761	45	34	[	[	X
ejde-761	45	35	23	23	NUM
ejde-761	45	36	]	]	NUM
ejde-761	45	37	)	)	PUNCT
ejde-761	45	38	,	,	PUNCT
ejde-761	45	39	where	where	SCONJ
ejde-761	45	40	m	m	NOUN
ejde-761	45	41	=	=	X
ejde-761	45	42	p1	p1	NOUN
ejde-761	45	43	−	−	PROPN
ejde-761	45	44	2	2	NUM
ejde-761	45	45	,	,	PUNCT
ejde-761	45	46	indeed	indeed	ADV
ejde-761	45	47	∆p1,2u	∆p1,2u	PUNCT
ejde-761	45	48	=	=	SYM
ejde-761	45	49	−∆(|∆u|p1−2∆u	−∆(|∆u|p1−2∆u	NOUN
ejde-761	45	50	)	)	PUNCT
ejde-761	45	51	.	.	PUNCT
ejde-761	46	1	(	(	PUNCT
ejde-761	46	2	1.2	1.2	NUM
ejde-761	46	3	)	)	PUNCT
ejde-761	46	4	as	as	SCONJ
ejde-761	46	5	described	describe	VERB
ejde-761	46	6	in	in	ADP
ejde-761	46	7	[	[	X
ejde-761	46	8	42	42	NUM
ejde-761	46	9	]	]	PUNCT
ejde-761	46	10	,	,	PUNCT
ejde-761	46	11	the	the	DET
ejde-761	46	12	p	p	NOUN
ejde-761	46	13	-	-	PUNCT
ejde-761	46	14	laplacian	laplacian	ADJ
ejde-761	46	15	higher	high	ADJ
ejde-761	46	16	order	order	NOUN
ejde-761	46	17	operator	operator	NOUN
ejde-761	46	18	preserves	preserve	VERB
ejde-761	46	19	the	the	DET
ejde-761	46	20	finite	finite	ADJ
ejde-761	46	21	propagation	propagation	NOUN
ejde-761	46	22	feature	feature	NOUN
ejde-761	46	23	from	from	ADP
ejde-761	46	24	the	the	DET
ejde-761	46	25	single	single	ADJ
ejde-761	46	26	p	p	NOUN
ejde-761	46	27	-	-	PUNCT
ejde-761	46	28	laplacian	laplacian	NOUN
ejde-761	46	29	.	.	PUNCT
ejde-761	47	1	in	in	ADP
ejde-761	47	2	addition	addition	NOUN
ejde-761	47	3	,	,	PUNCT
ejde-761	47	4	we	we	PRON
ejde-761	47	5	mention	mention	VERB
ejde-761	47	6	that	that	SCONJ
ejde-761	47	7	a	a	DET
ejde-761	47	8	space	space	NOUN
ejde-761	47	9	of	of	ADP
ejde-761	47	10	functions	function	NOUN
ejde-761	47	11	hn	hn	PROPN
ejde-761	47	12	0	0	PROPN
ejde-761	47	13	is	be	AUX
ejde-761	47	14	introduced	introduce	VERB
ejde-761	47	15	in	in	ADP
ejde-761	47	16	(	(	PUNCT
ejde-761	47	17	1.1	1.1	NUM
ejde-761	47	18	)	)	PUNCT
ejde-761	47	19	to	to	PART
ejde-761	47	20	provide	provide	VERB
ejde-761	47	21	a	a	DET
ejde-761	47	22	mollifying	mollifying	ADJ
ejde-761	47	23	space	space	NOUN
ejde-761	47	24	(	(	PUNCT
ejde-761	47	25	this	this	PRON
ejde-761	47	26	will	will	AUX
ejde-761	47	27	be	be	AUX
ejde-761	47	28	further	far	ADV
ejde-761	47	29	defined	define	VERB
ejde-761	47	30	afterward	afterward	ADV
ejde-761	47	31	)	)	PUNCT
ejde-761	47	32	to	to	PART
ejde-761	47	33	account	account	VERB
ejde-761	47	34	for	for	ADP
ejde-761	47	35	potential	potential	ADJ
ejde-761	47	36	higher	high	ADJ
ejde-761	47	37	order	order	NOUN
ejde-761	47	38	instabilities	instability	NOUN
ejde-761	47	39	close	close	VERB
ejde-761	47	40	the	the	DET
ejde-761	47	41	null	null	ADJ
ejde-761	47	42	critical	critical	ADJ
ejde-761	47	43	point	point	NOUN
ejde-761	47	44	.	.	PUNCT
ejde-761	48	1	the	the	DET
ejde-761	48	2	analysis	analysis	NOUN
ejde-761	48	3	of	of	ADP
ejde-761	48	4	problem	problem	NOUN
ejde-761	48	5	(	(	PUNCT
ejde-761	48	6	1.1	1.1	NUM
ejde-761	48	7	)	)	PUNCT
ejde-761	48	8	begins	begin	VERB
ejde-761	48	9	with	with	ADP
ejde-761	48	10	the	the	DET
ejde-761	48	11	definition	definition	NOUN
ejde-761	48	12	of	of	ADP
ejde-761	48	13	energy	energy	NOUN
ejde-761	48	14	solutions	solution	NOUN
ejde-761	48	15	,	,	PUNCT
ejde-761	48	16	as	as	SCONJ
ejde-761	48	17	proposed	propose	VERB
ejde-761	48	18	for	for	ADP
ejde-761	48	19	general	general	ADJ
ejde-761	48	20	diffusion	diffusion	NOUN
ejde-761	48	21	in	in	ADP
ejde-761	48	22	[	[	X
ejde-761	48	23	26	26	NUM
ejde-761	48	24	]	]	PUNCT
ejde-761	48	25	.	.	PUNCT
ejde-761	49	1	the	the	DET
ejde-761	49	2	problem	problem	NOUN
ejde-761	49	3	is	be	AUX
ejde-761	49	4	subsequently	subsequently	ADV
ejde-761	49	5	explored	explore	VERB
ejde-761	49	6	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	49	7	instability	instability	NOUN
ejde-761	49	8	of	of	ADP
ejde-761	49	9	energy	energy	NOUN
ejde-761	49	10	solutions	solution	NOUN
ejde-761	49	11	3	3	NUM
ejde-761	49	12	using	use	VERB
ejde-761	49	13	traveling	travel	VERB
ejde-761	49	14	wave	wave	NOUN
ejde-761	49	15	(	(	PUNCT
ejde-761	49	16	tw	tw	NOUN
ejde-761	49	17	)	)	PUNCT
ejde-761	49	18	solutions	solution	NOUN
ejde-761	49	19	,	,	PUNCT
ejde-761	49	20	an	an	DET
ejde-761	49	21	approach	approach	NOUN
ejde-761	49	22	introduced	introduce	VERB
ejde-761	49	23	in	in	ADP
ejde-761	49	24	the	the	DET
ejde-761	49	25	1930s	1930s	NUM
ejde-761	49	26	by	by	ADP
ejde-761	49	27	fisher	fisher	PROPN
ejde-761	50	1	[	[	X
ejde-761	50	2	22	22	NUM
ejde-761	50	3	]	]	PUNCT
ejde-761	50	4	and	and	CCONJ
ejde-761	50	5	kolmogorov	kolmogorov	ADJ
ejde-761	50	6	,	,	PUNCT
ejde-761	50	7	petrovskii	petrovskii	ADJ
ejde-761	50	8	,	,	PUNCT
ejde-761	50	9	and	and	CCONJ
ejde-761	50	10	piskunov	piskunov	NOUN
ejde-761	50	11	[	[	X
ejde-761	50	12	32	32	NUM
ejde-761	50	13	]	]	PUNCT
ejde-761	50	14	.	.	PUNCT
ejde-761	51	1	both	both	DET
ejde-761	51	2	studies	study	NOUN
ejde-761	51	3	were	be	AUX
ejde-761	51	4	grounded	ground	VERB
ejde-761	51	5	in	in	ADP
ejde-761	51	6	second	second	ADJ
ejde-761	51	7	-	-	PUNCT
ejde-761	51	8	order	order	NOUN
ejde-761	51	9	diffusion	diffusion	NOUN
ejde-761	51	10	and	and	CCONJ
ejde-761	51	11	a	a	DET
ejde-761	51	12	bi	bi	ADJ
ejde-761	51	13	-	-	ADJ
ejde-761	51	14	stable	stable	ADJ
ejde-761	51	15	nonlinear	nonlinear	ADJ
ejde-761	51	16	reaction	reaction	NOUN
ejde-761	51	17	term	term	NOUN
ejde-761	51	18	of	of	ADP
ejde-761	51	19	the	the	DET
ejde-761	51	20	form	form	NOUN
ejde-761	51	21	f(u	f(u	PROPN
ejde-761	51	22	)	)	PUNCT
ejde-761	52	1	=	=	PUNCT
ejde-761	52	2	u(1−	u(1−	PROPN
ejde-761	52	3	u	u	NOUN
ejde-761	52	4	)	)	PUNCT
ejde-761	52	5	.	.	PUNCT
ejde-761	53	1	the	the	DET
ejde-761	53	2	primary	primary	ADJ
ejde-761	53	3	question	question	NOUN
ejde-761	53	4	they	they	PRON
ejde-761	53	5	addressed	address	VERB
ejde-761	53	6	was	be	AUX
ejde-761	53	7	the	the	DET
ejde-761	53	8	existence	existence	NOUN
ejde-761	53	9	of	of	ADP
ejde-761	53	10	a	a	DET
ejde-761	53	11	tw	tw	NOUN
ejde-761	53	12	velocity	velocity	NOUN
ejde-761	53	13	that	that	PRON
ejde-761	53	14	produces	produce	VERB
ejde-761	53	15	a	a	DET
ejde-761	53	16	monotone	monotone	ADJ
ejde-761	53	17	front	front	ADJ
ejde-761	53	18	profile	profile	NOUN
ejde-761	53	19	,	,	PUNCT
ejde-761	53	20	free	free	ADJ
ejde-761	53	21	from	from	ADP
ejde-761	53	22	oscillations	oscillation	NOUN
ejde-761	53	23	.	.	PUNCT
ejde-761	54	1	since	since	SCONJ
ejde-761	54	2	its	its	PRON
ejde-761	54	3	inception	inception	NOUN
ejde-761	54	4	,	,	PUNCT
ejde-761	54	5	the	the	DET
ejde-761	54	6	fisher	fisher	PROPN
ejde-761	54	7	-	-	PUNCT
ejde-761	54	8	kpp	kpp	PROPN
ejde-761	54	9	model	model	NOUN
ejde-761	54	10	has	have	AUX
ejde-761	54	11	garnered	garner	VERB
ejde-761	54	12	significant	significant	ADJ
ejde-761	54	13	attention	attention	NOUN
ejde-761	54	14	,	,	PUNCT
ejde-761	54	15	with	with	ADP
ejde-761	54	16	applications	application	NOUN
ejde-761	54	17	extending	extend	VERB
ejde-761	54	18	to	to	PART
ejde-761	54	19	diverse	diverse	ADJ
ejde-761	54	20	fields	field	NOUN
ejde-761	54	21	such	such	ADJ
ejde-761	54	22	as	as	ADP
ejde-761	54	23	ecology	ecology	NOUN
ejde-761	54	24	,	,	PUNCT
ejde-761	54	25	biology	biology	NOUN
ejde-761	54	26	(	(	PUNCT
ejde-761	54	27	see	see	VERB
ejde-761	54	28	[	[	X
ejde-761	54	29	4	4	NUM
ejde-761	54	30	,	,	PUNCT
ejde-761	54	31	5	5	NUM
ejde-761	54	32	,	,	PUNCT
ejde-761	54	33	6	6	NUM
ejde-761	54	34	]	]	NUM
ejde-761	54	35	)	)	PUNCT
ejde-761	54	36	,	,	PUNCT
ejde-761	54	37	and	and	CCONJ
ejde-761	54	38	non	non	ADJ
ejde-761	54	39	-	-	ADJ
ejde-761	54	40	newtonian	newtonian	ADJ
ejde-761	54	41	fluids	fluid	NOUN
ejde-761	54	42	[	[	X
ejde-761	54	43	33	33	NUM
ejde-761	54	44	]	]	PUNCT
ejde-761	54	45	.	.	PUNCT
ejde-761	55	1	furthermore	furthermore	ADV
ejde-761	55	2	,	,	PUNCT
ejde-761	55	3	various	various	ADJ
ejde-761	55	4	bi	bi	ADJ
ejde-761	55	5	-	-	ADJ
ejde-761	55	6	stable	stable	ADJ
ejde-761	55	7	equations	equation	NOUN
ejde-761	55	8	have	have	AUX
ejde-761	55	9	been	be	AUX
ejde-761	55	10	analyzed	analyze	VERB
ejde-761	55	11	similarly	similarly	ADV
ejde-761	55	12	to	to	ADP
ejde-761	55	13	the	the	DET
ejde-761	55	14	fisher	fisher	PROPN
ejde-761	55	15	-	-	PUNCT
ejde-761	55	16	kpp	kpp	PROPN
ejde-761	55	17	model	model	NOUN
ejde-761	55	18	(	(	PUNCT
ejde-761	55	19	see	see	VERB
ejde-761	55	20	[	[	X
ejde-761	55	21	41	41	NUM
ejde-761	55	22	]	]	PUNCT
ejde-761	55	23	and	and	CCONJ
ejde-761	55	24	references	reference	NOUN
ejde-761	55	25	therein	therein	ADV
ejde-761	55	26	)	)	PUNCT
ejde-761	55	27	.	.	PUNCT
ejde-761	56	1	in	in	ADP
ejde-761	56	2	some	some	DET
ejde-761	56	3	cases	case	NOUN
ejde-761	56	4	,	,	PUNCT
ejde-761	56	5	these	these	DET
ejde-761	56	6	equations	equation	NOUN
ejde-761	56	7	have	have	AUX
ejde-761	56	8	been	be	AUX
ejde-761	56	9	extended	extend	VERB
ejde-761	56	10	to	to	PART
ejde-761	56	11	incorporate	incorporate	VERB
ejde-761	56	12	non	non	ADJ
ejde-761	56	13	-	-	ADJ
ejde-761	56	14	homogeneous	homogeneous	ADJ
ejde-761	56	15	diffusion	diffusion	NOUN
ejde-761	56	16	characterized	characterize	VERB
ejde-761	56	17	by	by	ADP
ejde-761	56	18	higher	high	ADJ
ejde-761	56	19	-	-	PUNCT
ejde-761	56	20	order	order	NOUN
ejde-761	56	21	diffusive	diffusive	ADJ
ejde-761	56	22	phenomena	phenomenon	NOUN
ejde-761	56	23	,	,	PUNCT
ejde-761	56	24	as	as	SCONJ
ejde-761	56	25	seen	see	VERB
ejde-761	56	26	in	in	ADP
ejde-761	56	27	the	the	DET
ejde-761	56	28	extended	extended	ADJ
ejde-761	56	29	fisher	fisher	PROPN
ejde-761	56	30	-	-	PUNCT
ejde-761	56	31	kolmogorov	kolmogorov	ADJ
ejde-761	56	32	equation	equation	NOUN
ejde-761	56	33	in	in	ADP
ejde-761	56	34	bi	bi	ADJ
ejde-761	56	35	-	-	ADJ
ejde-761	56	36	stable	stable	ADJ
ejde-761	56	37	systems	system	NOUN
ejde-761	56	38	[	[	X
ejde-761	56	39	11	11	NUM
ejde-761	56	40	,	,	PUNCT
ejde-761	56	41	16	16	NUM
ejde-761	56	42	,	,	PUNCT
ejde-761	56	43	39	39	NUM
ejde-761	56	44	]	]	PUNCT
ejde-761	56	45	.	.	PUNCT
ejde-761	57	1	additionally	additionally	ADV
ejde-761	57	2	,	,	PUNCT
ejde-761	57	3	a	a	DET
ejde-761	57	4	p	p	ADJ
ejde-761	57	5	-	-	PUNCT
ejde-761	57	6	laplacian	laplacian	ADJ
ejde-761	57	7	operator	operator	NOUN
ejde-761	57	8	has	have	AUX
ejde-761	57	9	been	be	AUX
ejde-761	57	10	introduced	introduce	VERB
ejde-761	57	11	to	to	PART
ejde-761	57	12	model	model	VERB
ejde-761	57	13	a	a	DET
ejde-761	57	14	fisher	fisher	PROPN
ejde-761	57	15	-	-	PUNCT
ejde-761	57	16	kpp	kpp	NOUN
ejde-761	57	17	type	type	NOUN
ejde-761	57	18	equation	equation	NOUN
ejde-761	57	19	in	in	ADP
ejde-761	57	20	[	[	X
ejde-761	57	21	7	7	NUM
ejde-761	57	22	]	]	PUNCT
ejde-761	57	23	.	.	PUNCT
ejde-761	58	1	tw	tw	NOUN
ejde-761	58	2	solutions	solution	NOUN
ejde-761	58	3	have	have	AUX
ejde-761	58	4	become	become	VERB
ejde-761	58	5	increasingly	increasingly	ADV
ejde-761	58	6	important	important	ADJ
ejde-761	58	7	in	in	ADP
ejde-761	58	8	applied	apply	VERB
ejde-761	58	9	sciences	science	NOUN
ejde-761	58	10	(	(	PUNCT
ejde-761	58	11	see	see	VERB
ejde-761	58	12	[	[	X
ejde-761	58	13	40	40	NUM
ejde-761	58	14	]	]	PUNCT
ejde-761	58	15	,	,	PUNCT
ejde-761	58	16	[	[	X
ejde-761	58	17	36	36	NUM
ejde-761	58	18	]	]	PUNCT
ejde-761	58	19	,	,	PUNCT
ejde-761	58	20	and	and	CCONJ
ejde-761	58	21	[	[	X
ejde-761	58	22	18	18	NUM
ejde-761	58	23	]	]	NUM
ejde-761	58	24	)	)	PUNCT
ejde-761	58	25	,	,	PUNCT
ejde-761	58	26	and	and	CCONJ
ejde-761	58	27	their	their	PRON
ejde-761	58	28	analysis	analysis	NOUN
ejde-761	58	29	has	have	AUX
ejde-761	58	30	been	be	AUX
ejde-761	58	31	extended	extend	VERB
ejde-761	58	32	to	to	ADP
ejde-761	58	33	the	the	DET
ejde-761	58	34	study	study	NOUN
ejde-761	58	35	of	of	ADP
ejde-761	58	36	higher	high	ADJ
ejde-761	58	37	-	-	PUNCT
ejde-761	58	38	order	order	NOUN
ejde-761	58	39	operators	operator	NOUN
ejde-761	58	40	(	(	PUNCT
ejde-761	58	41	see	see	VERB
ejde-761	58	42	[	[	X
ejde-761	58	43	28	28	NUM
ejde-761	58	44	,	,	PUNCT
ejde-761	58	45	34	34	NUM
ejde-761	58	46	,	,	PUNCT
ejde-761	58	47	24	24	NUM
ejde-761	58	48	,	,	PUNCT
ejde-761	58	49	25	25	NUM
ejde-761	58	50	]	]	PUNCT
ejde-761	58	51	)	)	PUNCT
ejde-761	58	52	.	.	PUNCT
ejde-761	59	1	this	this	DET
ejde-761	59	2	study	study	NOUN
ejde-761	59	3	aims	aim	VERB
ejde-761	59	4	to	to	PART
ejde-761	59	5	analyze	analyze	VERB
ejde-761	59	6	the	the	DET
ejde-761	59	7	oscillatory	oscillatory	ADJ
ejde-761	59	8	profiles	profile	NOUN
ejde-761	59	9	of	of	ADP
ejde-761	59	10	tw	tw	NOUN
ejde-761	59	11	solutions	solution	NOUN
ejde-761	59	12	using	use	VERB
ejde-761	59	13	both	both	CCONJ
ejde-761	59	14	analytical	analytical	ADJ
ejde-761	59	15	and	and	CCONJ
ejde-761	59	16	numerical	numerical	ADJ
ejde-761	59	17	approaches	approach	NOUN
ejde-761	59	18	,	,	PUNCT
ejde-761	59	19	the	the	DET
ejde-761	59	20	latter	latter	ADJ
ejde-761	59	21	serving	serve	VERB
ejde-761	59	22	to	to	PART
ejde-761	59	23	validate	validate	VERB
ejde-761	59	24	the	the	DET
ejde-761	59	25	analytical	analytical	ADJ
ejde-761	59	26	results	result	NOUN
ejde-761	59	27	.	.	PUNCT
ejde-761	60	1	the	the	DET
ejde-761	60	2	analysis	analysis	NOUN
ejde-761	60	3	is	be	AUX
ejde-761	60	4	based	base	VERB
ejde-761	60	5	on	on	ADP
ejde-761	60	6	the	the	DET
ejde-761	60	7	energy	energy	NOUN
ejde-761	60	8	formulation	formulation	NOUN
ejde-761	60	9	introduced	introduce	VERB
ejde-761	60	10	in	in	ADP
ejde-761	60	11	[	[	X
ejde-761	60	12	26	26	NUM
ejde-761	60	13	]	]	PUNCT
ejde-761	60	14	,	,	PUNCT
ejde-761	60	15	and	and	CCONJ
ejde-761	60	16	it	it	PRON
ejde-761	60	17	employs	employ	VERB
ejde-761	60	18	functional	functional	ADJ
ejde-761	60	19	spaces	space	NOUN
ejde-761	60	20	to	to	PART
ejde-761	60	21	represent	represent	VERB
ejde-761	60	22	the	the	DET
ejde-761	60	23	properties	property	NOUN
ejde-761	60	24	of	of	ADP
ejde-761	60	25	solutions	solution	NOUN
ejde-761	60	26	.	.	PUNCT
ejde-761	61	1	in	in	ADP
ejde-761	61	2	particular	particular	ADJ
ejde-761	61	3	,	,	PUNCT
ejde-761	61	4	generalized	generalized	ADJ
ejde-761	61	5	sobolev	sobolev	NOUN
ejde-761	61	6	spaces	space	NOUN
ejde-761	61	7	are	be	AUX
ejde-761	61	8	utilized	utilize	VERB
ejde-761	61	9	to	to	PART
ejde-761	61	10	account	account	VERB
ejde-761	61	11	for	for	ADP
ejde-761	61	12	compact	compact	ADJ
ejde-761	61	13	support	support	NOUN
ejde-761	61	14	functions	function	NOUN
ejde-761	61	15	,	,	PUNCT
ejde-761	61	16	mollifiers	mollifier	NOUN
ejde-761	61	17	,	,	PUNCT
ejde-761	61	18	and	and	CCONJ
ejde-761	61	19	oscillatory	oscillatory	ADJ
ejde-761	61	20	patterns	pattern	NOUN
ejde-761	61	21	.	.	PUNCT
ejde-761	62	1	problem	problem	NOUN
ejde-761	62	2	(	(	PUNCT
ejde-761	62	3	1.1	1.1	NUM
ejde-761	62	4	)	)	PUNCT
ejde-761	62	5	is	be	AUX
ejde-761	62	6	then	then	ADV
ejde-761	62	7	reformulated	reformulate	VERB
ejde-761	62	8	in	in	ADP
ejde-761	62	9	the	the	DET
ejde-761	62	10	tw	tw	NOUN
ejde-761	62	11	framework	framework	NOUN
ejde-761	62	12	,	,	PUNCT
ejde-761	62	13	where	where	SCONJ
ejde-761	62	14	the	the	DET
ejde-761	62	15	oscillatory	oscillatory	ADJ
ejde-761	62	16	properties	property	NOUN
ejde-761	62	17	of	of	ADP
ejde-761	62	18	solutions	solution	NOUN
ejde-761	62	19	—	—	PUNCT
ejde-761	62	20	often	often	ADV
ejde-761	62	21	referred	refer	VERB
ejde-761	62	22	to	to	ADP
ejde-761	62	23	as	as	ADP
ejde-761	62	24	solution	solution	NOUN
ejde-761	62	25	instabilities	instability	NOUN
ejde-761	62	26	—	—	PUNCT
ejde-761	62	27	are	be	AUX
ejde-761	62	28	examined	examine	VERB
ejde-761	62	29	.	.	PUNCT
ejde-761	63	1	the	the	DET
ejde-761	63	2	spectrum	spectrum	NOUN
ejde-761	63	3	of	of	ADP
ejde-761	63	4	the	the	DET
ejde-761	63	5	higher	high	ADJ
ejde-761	63	6	-	-	PUNCT
ejde-761	63	7	order	order	NOUN
ejde-761	63	8	p	p	ADJ
ejde-761	63	9	-	-	PUNCT
ejde-761	63	10	laplacian	laplacian	ADJ
ejde-761	63	11	operator	operator	NOUN
ejde-761	63	12	is	be	AUX
ejde-761	63	13	explored	explore	VERB
ejde-761	63	14	using	use	VERB
ejde-761	63	15	a	a	DET
ejde-761	63	16	homotopy	homotopy	NOUN
ejde-761	63	17	representation	representation	NOUN
ejde-761	63	18	near	near	ADP
ejde-761	63	19	the	the	DET
ejde-761	63	20	null	null	ADJ
ejde-761	63	21	solution	solution	NOUN
ejde-761	63	22	.	.	PUNCT
ejde-761	64	1	finally	finally	ADV
ejde-761	64	2	,	,	PUNCT
ejde-761	64	3	a	a	DET
ejde-761	64	4	numerical	numerical	ADJ
ejde-761	64	5	approach	approach	NOUN
ejde-761	64	6	confirms	confirm	VERB
ejde-761	64	7	the	the	DET
ejde-761	64	8	attractive	attractive	ADJ
ejde-761	64	9	properties	property	NOUN
ejde-761	64	10	of	of	ADP
ejde-761	64	11	the	the	DET
ejde-761	64	12	null	null	ADJ
ejde-761	64	13	solution	solution	NOUN
ejde-761	64	14	and	and	CCONJ
ejde-761	64	15	the	the	DET
ejde-761	64	16	stabilizing	stabilize	VERB
ejde-761	64	17	effect	effect	NOUN
ejde-761	64	18	of	of	ADP
ejde-761	64	19	this	this	DET
ejde-761	64	20	null	null	ADJ
ejde-761	64	21	solution	solution	NOUN
ejde-761	64	22	as	as	ADP
ejde-761	64	23	the	the	DET
ejde-761	64	24	tw	tw	NOUN
ejde-761	64	25	speed	speed	NOUN
ejde-761	64	26	increases	increase	NOUN
ejde-761	64	27	.	.	PUNCT
ejde-761	65	1	it	it	PRON
ejde-761	65	2	is	be	AUX
ejde-761	65	3	important	important	ADJ
ejde-761	65	4	to	to	PART
ejde-761	65	5	note	note	VERB
ejde-761	65	6	that	that	SCONJ
ejde-761	65	7	the	the	DET
ejde-761	65	8	mathematical	mathematical	ADJ
ejde-761	65	9	treatment	treatment	NOUN
ejde-761	65	10	in	in	ADP
ejde-761	65	11	the	the	DET
ejde-761	65	12	tw	tw	NOUN
ejde-761	65	13	domain	domain	NOUN
ejde-761	65	14	begins	begin	VERB
ejde-761	65	15	with	with	ADP
ejde-761	65	16	step	step	NOUN
ejde-761	65	17	-	-	PUNCT
ejde-761	65	18	like	like	ADJ
ejde-761	65	19	initial	initial	ADJ
ejde-761	65	20	data	datum	NOUN
ejde-761	65	21	.	.	PUNCT
ejde-761	66	1	although	although	SCONJ
ejde-761	66	2	this	this	DET
ejde-761	66	3	step	step	NOUN
ejde-761	66	4	function	function	NOUN
ejde-761	66	5	is	be	AUX
ejde-761	66	6	not	not	PART
ejde-761	66	7	compactly	compactly	ADV
ejde-761	66	8	supported	support	VERB
ejde-761	66	9	,	,	PUNCT
ejde-761	66	10	as	as	SCONJ
ejde-761	66	11	required	require	VERB
ejde-761	66	12	by	by	ADP
ejde-761	66	13	problem	problem	NOUN
ejde-761	66	14	(	(	PUNCT
ejde-761	66	15	1.1	1.1	NUM
ejde-761	66	16	)	)	PUNCT
ejde-761	66	17	,	,	PUNCT
ejde-761	66	18	it	it	PRON
ejde-761	66	19	is	be	AUX
ejde-761	66	20	assumed	assume	VERB
ejde-761	66	21	that	that	SCONJ
ejde-761	66	22	such	such	DET
ejde-761	66	23	an	an	DET
ejde-761	66	24	initial	initial	ADJ
ejde-761	66	25	function	function	NOUN
ejde-761	66	26	provides	provide	VERB
ejde-761	66	27	a	a	DET
ejde-761	66	28	suitable	suitable	ADJ
ejde-761	66	29	starting	starting	NOUN
ejde-761	66	30	condition	condition	NOUN
ejde-761	66	31	to	to	PART
ejde-761	66	32	study	study	VERB
ejde-761	66	33	the	the	DET
ejde-761	66	34	evolution	evolution	NOUN
ejde-761	66	35	of	of	ADP
ejde-761	66	36	a	a	DET
ejde-761	66	37	positive	positive	ADJ
ejde-761	66	38	mass	mass	NOUN
ejde-761	66	39	alongside	alongside	ADP
ejde-761	66	40	a	a	DET
ejde-761	66	41	null	null	ADJ
ejde-761	66	42	state	state	NOUN
ejde-761	66	43	.	.	PUNCT
ejde-761	67	1	the	the	DET
ejde-761	67	2	methodology	methodology	NOUN
ejde-761	67	3	employed	employ	VERB
ejde-761	67	4	in	in	ADP
ejde-761	67	5	this	this	DET
ejde-761	67	6	work	work	NOUN
ejde-761	67	7	combines	combine	VERB
ejde-761	67	8	an	an	DET
ejde-761	67	9	analytical	analytical	ADJ
ejde-761	67	10	approach	approach	NOUN
ejde-761	67	11	to	to	PART
ejde-761	67	12	demonstrate	demonstrate	VERB
ejde-761	67	13	the	the	DET
ejde-761	67	14	instabilities	instability	NOUN
ejde-761	67	15	of	of	ADP
ejde-761	67	16	the	the	DET
ejde-761	67	17	traveling	travel	VERB
ejde-761	67	18	wave	wave	NOUN
ejde-761	67	19	(	(	PUNCT
ejde-761	67	20	tw	tw	NOUN
ejde-761	67	21	)	)	PUNCT
ejde-761	67	22	profiles	profile	NOUN
ejde-761	67	23	with	with	ADP
ejde-761	67	24	a	a	DET
ejde-761	67	25	numerical	numerical	ADJ
ejde-761	67	26	validation	validation	NOUN
ejde-761	67	27	using	use	VERB
ejde-761	67	28	the	the	DET
ejde-761	67	29	bvp4c	bvp4c	PROPN
ejde-761	67	30	function	function	NOUN
ejde-761	67	31	in	in	ADP
ejde-761	67	32	matlab	matlab	PROPN
ejde-761	67	33	to	to	PART
ejde-761	67	34	support	support	VERB
ejde-761	67	35	the	the	DET
ejde-761	67	36	analysis	analysis	NOUN
ejde-761	67	37	.	.	PUNCT
ejde-761	68	1	it	it	PRON
ejde-761	68	2	is	be	AUX
ejde-761	68	3	important	important	ADJ
ejde-761	68	4	to	to	PART
ejde-761	68	5	note	note	VERB
ejde-761	68	6	that	that	SCONJ
ejde-761	68	7	the	the	DET
ejde-761	68	8	analysis	analysis	NOUN
ejde-761	68	9	introduced	introduce	VERB
ejde-761	68	10	by	by	ADP
ejde-761	68	11	galaktionov	galaktionov	PROPN
ejde-761	68	12	in	in	ADP
ejde-761	68	13	[	[	X
ejde-761	68	14	23	23	NUM
ejde-761	68	15	]	]	PUNCT
ejde-761	68	16	discusses	discuss	VERB
ejde-761	68	17	blow	blow	NOUN
ejde-761	68	18	-	-	PUNCT
ejde-761	68	19	up	up	ADP
ejde-761	68	20	profiles	profile	NOUN
ejde-761	68	21	,	,	PUNCT
ejde-761	68	22	yielding	yield	VERB
ejde-761	68	23	exact	exact	ADJ
ejde-761	68	24	patterns	pattern	NOUN
ejde-761	68	25	.	.	PUNCT
ejde-761	69	1	in	in	ADP
ejde-761	69	2	the	the	DET
ejde-761	69	3	present	present	ADJ
ejde-761	69	4	analysis	analysis	NOUN
ejde-761	69	5	,	,	PUNCT
ejde-761	69	6	we	we	PRON
ejde-761	69	7	provide	provide	VERB
ejde-761	69	8	evidence	evidence	NOUN
ejde-761	69	9	of	of	ADP
ejde-761	69	10	instabilities	instability	NOUN
ejde-761	69	11	in	in	ADP
ejde-761	69	12	the	the	DET
ejde-761	69	13	tw	tw	PROPN
ejde-761	69	14	solutions	solution	NOUN
ejde-761	69	15	prior	prior	ADV
ejde-761	69	16	to	to	ADP
ejde-761	69	17	the	the	DET
ejde-761	69	18	onset	onset	NOUN
ejde-761	69	19	of	of	ADP
ejde-761	69	20	blow	blow	NOUN
ejde-761	69	21	-	-	PUNCT
ejde-761	69	22	up	up	NOUN
ejde-761	69	23	.	.	PUNCT
ejde-761	70	1	additionally	additionally	ADV
ejde-761	70	2	,	,	PUNCT
ejde-761	70	3	we	we	PRON
ejde-761	70	4	characterize	characterize	VERB
ejde-761	70	5	the	the	DET
ejde-761	70	6	null	null	ADJ
ejde-761	70	7	solution	solution	NOUN
ejde-761	70	8	as	as	ADP
ejde-761	70	9	an	an	DET
ejde-761	70	10	attractor	attractor	NOUN
ejde-761	70	11	,	,	PUNCT
ejde-761	70	12	which	which	PRON
ejde-761	70	13	significantly	significantly	ADV
ejde-761	70	14	influences	influence	VERB
ejde-761	70	15	the	the	DET
ejde-761	70	16	single	single	ADJ
ejde-761	70	17	-	-	PUNCT
ejde-761	70	18	point	point	NOUN
ejde-761	70	19	blow	blow	NOUN
ejde-761	70	20	-	-	PUNCT
ejde-761	70	21	up	up	ADP
ejde-761	70	22	behavior	behavior	NOUN
ejde-761	70	23	.	.	PUNCT
ejde-761	71	1	specifically	specifically	ADV
ejde-761	71	2	,	,	PUNCT
ejde-761	71	3	while	while	SCONJ
ejde-761	71	4	solutions	solution	NOUN
ejde-761	71	5	exhibit	exhibit	VERB
ejde-761	71	6	oscillations	oscillation	NOUN
ejde-761	71	7	as	as	SCONJ
ejde-761	71	8	they	they	PRON
ejde-761	71	9	approach	approach	VERB
ejde-761	71	10	the	the	DET
ejde-761	71	11	null	null	ADJ
ejde-761	71	12	solution	solution	NOUN
ejde-761	71	13	,	,	PUNCT
ejde-761	71	14	this	this	DET
ejde-761	71	15	state	state	NOUN
ejde-761	71	16	acts	act	VERB
ejde-761	71	17	as	as	ADP
ejde-761	71	18	an	an	DET
ejde-761	71	19	attractor	attractor	NOUN
ejde-761	71	20	,	,	PUNCT
ejde-761	71	21	thereby	thereby	ADV
ejde-761	71	22	preventing	prevent	VERB
ejde-761	71	23	the	the	DET
ejde-761	71	24	formation	formation	NOUN
ejde-761	71	25	of	of	ADP
ejde-761	71	26	blow	blow	NOUN
ejde-761	71	27	-	-	PUNCT
ejde-761	71	28	up	up	ADP
ejde-761	71	29	patterns	pattern	NOUN
ejde-761	71	30	.	.	PUNCT
ejde-761	72	1	2	2	X
ejde-761	72	2	.	.	X
ejde-761	72	3	preliminaries	preliminary	NOUN
ejde-761	72	4	firstly	firstly	ADV
ejde-761	72	5	,	,	PUNCT
ejde-761	72	6	we	we	PRON
ejde-761	72	7	consider	consider	VERB
ejde-761	72	8	the	the	DET
ejde-761	72	9	definition	definition	NOUN
ejde-761	72	10	of	of	ADP
ejde-761	72	11	a	a	DET
ejde-761	72	12	generalized	generalized	ADJ
ejde-761	72	13	energy	energy	NOUN
ejde-761	72	14	solution	solution	NOUN
ejde-761	72	15	.	.	PUNCT
ejde-761	73	1	4	4	NUM
ejde-761	73	2	j.	j.	PROPN
ejde-761	73	3	l.	l.	PROPN
ejde-761	73	4	díaz	díaz	PROPN
ejde-761	73	5	palencia	palencia	PROPN
ejde-761	73	6	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	73	7	definition	definition	NOUN
ejde-761	73	8	2.1	2.1	NUM
ejde-761	73	9	.	.	PUNCT
ejde-761	74	1	u(x	u(x	PROPN
ejde-761	74	2	,	,	PUNCT
ejde-761	74	3	t	t	PROPN
ejde-761	74	4	)	)	PUNCT
ejde-761	74	5	is	be	AUX
ejde-761	74	6	said	say	VERB
ejde-761	74	7	to	to	PART
ejde-761	74	8	be	be	AUX
ejde-761	74	9	an	an	DET
ejde-761	74	10	energy	energy	NOUN
ejde-761	74	11	solution	solution	NOUN
ejde-761	74	12	to	to	ADP
ejde-761	74	13	problem	problem	NOUN
ejde-761	74	14	(	(	PUNCT
ejde-761	74	15	1.1	1.1	NUM
ejde-761	74	16	)	)	PUNCT
ejde-761	74	17	in	in	ADP
ejde-761	74	18	(	(	PUNCT
ejde-761	74	19	0	0	NUM
ejde-761	74	20	,	,	PUNCT
ejde-761	74	21	t	t	PROPN
ejde-761	74	22	)	)	PUNCT
ejde-761	74	23	if	if	SCONJ
ejde-761	74	24	for	for	ADP
ejde-761	74	25	any	any	DET
ejde-761	74	26	0	0	PUNCT
ejde-761	74	27	<	<	X
ejde-761	74	28	τ	τ	X
ejde-761	74	29	<	<	X
ejde-761	74	30	t	t	PROPN
ejde-761	74	31	,	,	PUNCT
ejde-761	74	32	the	the	DET
ejde-761	74	33	following	follow	VERB
ejde-761	74	34	holds	hold	VERB
ejde-761	74	35	(	(	PUNCT
ejde-761	74	36	see	see	VERB
ejde-761	74	37	[	[	X
ejde-761	74	38	26])∫	26])∫	NUM
ejde-761	74	39	τ	τ	X
ejde-761	74	40	0	0	NUM
ejde-761	74	41	∫	∫	PROPN
ejde-761	74	42	rn	rn	PROPN
ejde-761	74	43	utξ	utξ	PROPN
ejde-761	74	44	,	,	PUNCT
ejde-761	74	45	dx	dx	PROPN
ejde-761	74	46	dt+	dt+	NOUN
ejde-761	74	47	∫	∫	PROPN
ejde-761	74	48	τ	τ	X
ejde-761	74	49	0	0	NUM
ejde-761	75	1	∫	∫	PROPN
ejde-761	75	2	rn	rn	PROPN
ejde-761	76	1	−|∆u|m∆u∆ξ	−|∆u|m∆u∆ξ	PROPN
ejde-761	76	2	dx	dx	PROPN
ejde-761	76	3	dt+	dt+	NOUN
ejde-761	76	4	∫	∫	PROPN
ejde-761	76	5	τ	τ	X
ejde-761	76	6	0	0	NUM
ejde-761	76	7	∫	∫	PROPN
ejde-761	76	8	rn	rn	PROPN
ejde-761	76	9	|u|p−1uξ	|u|p−1uξ	PROPN
ejde-761	76	10	dx	dx	PROPN
ejde-761	76	11	dt	dt	NOUN
ejde-761	76	12	=	=	SYM
ejde-761	76	13	0	0	PROPN
ejde-761	76	14	,	,	PUNCT
ejde-761	76	15	(	(	PUNCT
ejde-761	76	16	2.1	2.1	NUM
ejde-761	76	17	)	)	PUNCT
ejde-761	76	18	for	for	ADP
ejde-761	76	19	any	any	DET
ejde-761	76	20	arbitrary	arbitrary	ADJ
ejde-761	76	21	function	function	NOUN
ejde-761	76	22	ξ	ξ	PROPN
ejde-761	76	23	∈	∈	PROPN
ejde-761	76	24	l1(0	l1(0	PROPN
ejde-761	76	25	,	,	PUNCT
ejde-761	76	26	τ	τ	PROPN
ejde-761	76	27	;	;	PUNCT
ejde-761	76	28	h	h	PROPN
ejde-761	76	29	2	2	NUM
ejde-761	76	30	0	0	NUM
ejde-761	76	31	(	(	PUNCT
ejde-761	76	32	rn	rn	PROPN
ejde-761	76	33	)	)	PUNCT
ejde-761	76	34	)	)	PUNCT
ejde-761	76	35	.	.	PUNCT
ejde-761	77	1	consider	consider	VERB
ejde-761	77	2	the	the	DET
ejde-761	77	3	following	follow	VERB
ejde-761	77	4	proposition	proposition	NOUN
ejde-761	77	5	based	base	VERB
ejde-761	77	6	on	on	ADP
ejde-761	77	7	results	result	NOUN
ejde-761	77	8	in	in	ADP
ejde-761	77	9	[	[	X
ejde-761	77	10	1	1	NUM
ejde-761	77	11	]	]	PUNCT
ejde-761	77	12	and	and	CCONJ
ejde-761	77	13	[	[	X
ejde-761	77	14	8	8	NUM
ejde-761	77	15	]	]	PUNCT
ejde-761	77	16	.	.	PUNCT
ejde-761	78	1	proposition	proposition	NOUN
ejde-761	78	2	2.2	2.2	NUM
ejde-761	78	3	.	.	PUNCT
ejde-761	79	1	let	let	VERB
ejde-761	79	2	f	f	X
ejde-761	79	3	,	,	PUNCT
ejde-761	79	4	g	g	PROPN
ejde-761	79	5	,	,	PUNCT
ejde-761	79	6	h	h	NOUN
ejde-761	79	7	∈	∈	PROPN
ejde-761	79	8	c	c	NOUN
ejde-761	79	9	n	n	X
ejde-761	79	10	0	0	NUM
ejde-761	79	11	(	(	PUNCT
ejde-761	79	12	rn	rn	PROPN
ejde-761	79	13	)	)	PUNCT
ejde-761	79	14	,	,	PUNCT
ejde-761	79	15	with	with	ADP
ejde-761	79	16	(	(	PUNCT
ejde-761	79	17	n	n	PRON
ejde-761	79	18	≥	≥	NOUN
ejde-761	79	19	1	1	NUM
ejde-761	79	20	)	)	PUNCT
ejde-761	79	21	,	,	PUNCT
ejde-761	79	22	the	the	DET
ejde-761	79	23	following	follow	VERB
ejde-761	79	24	anisotropic	anisotropic	NOUN
ejde-761	79	25	sobolev	sobolev	PROPN
ejde-761	79	26	inequality	inequality	PROPN
ejde-761	79	27	holds,∫	holds,∫	NOUN
ejde-761	79	28	rn	rn	PROPN
ejde-761	79	29	|fgh|	|fgh|	PROPN
ejde-761	79	30	dx	dx	PROPN
ejde-761	79	31	≤	≤	PROPN
ejde-761	79	32	k∥f∥	k∥f∥	PROPN
ejde-761	79	33	α−1	α−1	PROPN
ejde-761	79	34	α	α	PROPN
ejde-761	79	35	lq	lq	ADP
ejde-761	79	36	∥∇f∥1	∥∇f∥1	NOUN
ejde-761	79	37	/	/	SYM
ejde-761	79	38	αls	αl	VERB
ejde-761	79	39	∥g∥	∥g∥	PROPN
ejde-761	80	1	α−2	α−2	PROPN
ejde-761	80	2	α	α	PROPN
ejde-761	80	3	l2	l2	NOUN
ejde-761	80	4	∥∇g∥1	∥∇g∥1	NOUN
ejde-761	80	5	/	/	SYM
ejde-761	80	6	αl2	αl2	NOUN
ejde-761	80	7	∥h∥l2	∥h∥l2	NOUN
ejde-761	80	8	,	,	PUNCT
ejde-761	80	9	(	(	PUNCT
ejde-761	80	10	2.2	2.2	NUM
ejde-761	80	11	)	)	PUNCT
ejde-761	80	12	where	where	SCONJ
ejde-761	80	13	k	k	PROPN
ejde-761	80	14	>	>	X
ejde-761	80	15	0	0	PROPN
ejde-761	80	16	,	,	PUNCT
ejde-761	80	17	α−1	α−1	PROPN
ejde-761	80	18	q	q	PROPN
ejde-761	80	19	+	+	NUM
ejde-761	80	20	1	1	NUM
ejde-761	80	21	s	s	NOUN
ejde-761	80	22	=	=	SYM
ejde-761	80	23	1	1	NUM
ejde-761	80	24	,	,	PUNCT
ejde-761	80	25	α	α	NOUN
ejde-761	80	26	>	>	X
ejde-761	80	27	2	2	NUM
ejde-761	80	28	,	,	PUNCT
ejde-761	80	29	and	and	CCONJ
ejde-761	80	30	1	1	NUM
ejde-761	80	31	≤	≤	NUM
ejde-761	80	32	q	q	NOUN
ejde-761	80	33	,	,	PUNCT
ejde-761	80	34	s	s	PART
ejde-761	80	35	<	<	X
ejde-761	80	36	∞.	∞.	PROPN
ejde-761	80	37	according	accord	VERB
ejde-761	80	38	to	to	ADP
ejde-761	80	39	[	[	X
ejde-761	80	40	26	26	NUM
ejde-761	80	41	]	]	PUNCT
ejde-761	80	42	,	,	PUNCT
ejde-761	80	43	the	the	DET
ejde-761	80	44	asymptotic	asymptotic	ADJ
ejde-761	80	45	behaviour	behaviour	NOUN
ejde-761	80	46	of	of	ADP
ejde-761	80	47	solutions	solution	NOUN
ejde-761	80	48	exhibiting	exhibit	VERB
ejde-761	80	49	blow	blow	NOUN
ejde-761	80	50	-	-	PUNCT
ejde-761	80	51	up	up	NOUN
ejde-761	80	52	depends	depend	VERB
ejde-761	80	53	on	on	ADP
ejde-761	80	54	the	the	DET
ejde-761	80	55	asymptotic	asymptotic	ADJ
ejde-761	80	56	behaviour	behaviour	NOUN
ejde-761	80	57	of	of	ADP
ejde-761	80	58	a	a	DET
ejde-761	80	59	rescaled	rescaled	ADJ
ejde-761	80	60	kernel	kernel	NOUN
ejde-761	80	61	for	for	ADP
ejde-761	80	62	a	a	DET
ejde-761	80	63	linear	linear	ADJ
ejde-761	80	64	fourth	fourth	ADJ
ejde-761	80	65	-	-	PUNCT
ejde-761	80	66	order	order	NOUN
ejde-761	80	67	parabolic	parabolic	NOUN
ejde-761	80	68	operator	operator	NOUN
ejde-761	80	69	of	of	ADP
ejde-761	80	70	the	the	DET
ejde-761	80	71	form	form	NOUN
ejde-761	80	72	ut	ut	PROPN
ejde-761	80	73	=	=	SYM
ejde-761	80	74	−∆2u	−∆2u	PROPN
ejde-761	80	75	.	.	PROPN
ejde-761	81	1	this	this	DET
ejde-761	81	2	rescaled	rescaled	ADJ
ejde-761	81	3	kernel	kernel	NOUN
ejde-761	81	4	exhibits	exhibit	VERB
ejde-761	81	5	an	an	DET
ejde-761	81	6	exponential	exponential	ADJ
ejde-761	81	7	behaviour	behaviour	NOUN
ejde-761	81	8	of	of	ADP
ejde-761	81	9	the	the	DET
ejde-761	81	10	form	form	NOUN
ejde-761	81	11	∼	∼	VERB
ejde-761	81	12	e−x4/3	e−x4/3	NOUN
ejde-761	81	13	.	.	PUNCT
ejde-761	82	1	in	in	ADP
ejde-761	82	2	[	[	X
ejde-761	82	3	26	26	NUM
ejde-761	82	4	]	]	PUNCT
ejde-761	82	5	,	,	PUNCT
ejde-761	82	6	it	it	PRON
ejde-761	82	7	is	be	AUX
ejde-761	82	8	shown	show	VERB
ejde-761	82	9	that	that	SCONJ
ejde-761	82	10	the	the	DET
ejde-761	82	11	asymptotic	asymptotic	ADJ
ejde-761	82	12	behaviour	behaviour	NOUN
ejde-761	82	13	of	of	ADP
ejde-761	82	14	any	any	DET
ejde-761	82	15	possible	possible	ADJ
ejde-761	82	16	blow	blow	NOUN
ejde-761	82	17	-	-	PUNCT
ejde-761	82	18	up	up	ADP
ejde-761	82	19	profile	profile	NOUN
ejde-761	82	20	for	for	SCONJ
ejde-761	82	21	the	the	DET
ejde-761	82	22	higher	high	ADJ
ejde-761	82	23	-	-	PUNCT
ejde-761	82	24	order	order	NOUN
ejde-761	82	25	p	p	ADJ
ejde-761	82	26	-	-	PUNCT
ejde-761	82	27	laplacian	laplacian	ADJ
ejde-761	82	28	operator	operator	NOUN
ejde-761	82	29	exhibits	exhibit	VERB
ejde-761	82	30	a	a	DET
ejde-761	82	31	similar	similar	ADJ
ejde-761	82	32	exponential	exponential	ADJ
ejde-761	82	33	profile	profile	NOUN
ejde-761	82	34	.	.	PUNCT
ejde-761	83	1	the	the	DET
ejde-761	83	2	asymptotic	asymptotic	ADJ
ejde-761	83	3	estimates	estimate	NOUN
ejde-761	83	4	lead	lead	VERB
ejde-761	83	5	to	to	ADP
ejde-761	83	6	the	the	DET
ejde-761	83	7	proposal	proposal	NOUN
ejde-761	83	8	of	of	ADP
ejde-761	83	9	bundles	bundle	NOUN
ejde-761	83	10	of	of	ADP
ejde-761	83	11	blow	blow	NOUN
ejde-761	83	12	-	-	PUNCT
ejde-761	83	13	up	up	ADP
ejde-761	83	14	profiles	profile	NOUN
ejde-761	83	15	following	follow	VERB
ejde-761	83	16	s	s	PRON
ejde-761	83	17	and	and	CCONJ
ejde-761	83	18	hs	hs	NOUN
ejde-761	83	19	regimes	regime	NOUN
ejde-761	83	20	.	.	PUNCT
ejde-761	84	1	as	as	SCONJ
ejde-761	84	2	our	our	PRON
ejde-761	84	3	intention	intention	NOUN
ejde-761	84	4	is	be	AUX
ejde-761	84	5	to	to	PART
ejde-761	84	6	characterize	characterize	VERB
ejde-761	84	7	oscillatory	oscillatory	ADJ
ejde-761	84	8	exponential	exponential	ADJ
ejde-761	84	9	bundles	bundle	NOUN
ejde-761	84	10	of	of	ADP
ejde-761	84	11	solutions	solution	NOUN
ejde-761	84	12	close	close	ADJ
ejde-761	84	13	to	to	ADP
ejde-761	84	14	the	the	DET
ejde-761	84	15	critical	critical	ADJ
ejde-761	84	16	points	point	NOUN
ejde-761	84	17	,	,	PUNCT
ejde-761	84	18	a	a	DET
ejde-761	84	19	norm	norm	NOUN
ejde-761	84	20	is	be	AUX
ejde-761	84	21	introduced	introduce	VERB
ejde-761	84	22	with	with	ADP
ejde-761	84	23	an	an	DET
ejde-761	84	24	appropriate	appropriate	ADJ
ejde-761	84	25	weight	weight	NOUN
ejde-761	84	26	to	to	PART
ejde-761	84	27	scale	scale	VERB
ejde-761	84	28	out	out	ADP
ejde-761	84	29	the	the	DET
ejde-761	84	30	oscillatory	oscillatory	ADJ
ejde-761	84	31	profiles	profile	NOUN
ejde-761	84	32	close	close	ADJ
ejde-761	84	33	to	to	ADP
ejde-761	84	34	the	the	DET
ejde-761	84	35	null	null	ADJ
ejde-761	84	36	condition	condition	NOUN
ejde-761	84	37	.	.	PUNCT
ejde-761	85	1	definition	definition	NOUN
ejde-761	85	2	2.3	2.3	NUM
ejde-761	85	3	.	.	PUNCT
ejde-761	86	1	let	let	VERB
ejde-761	86	2	∥h∥2θ	∥h∥2θ	PROPN
ejde-761	86	3	=	=	SYM
ejde-761	86	4	∫	∫	PROPN
ejde-761	86	5	rn	rn	PROPN
ejde-761	86	6	θ(z	θ(z	PROPN
ejde-761	86	7	)	)	PUNCT
ejde-761	86	8	4∑	4∑	NOUN
ejde-761	86	9	j=0	j=0	VERB
ejde-761	86	10	|djh(z)|2dz	|djh(z)|2dz	NOUN
ejde-761	86	11	,	,	PUNCT
ejde-761	86	12	(	(	PUNCT
ejde-761	86	13	2.3	2.3	NUM
ejde-761	86	14	)	)	PUNCT
ejde-761	86	15	where	where	SCONJ
ejde-761	86	16	d	d	NOUN
ejde-761	86	17	=	=	SYM
ejde-761	86	18	d	d	X
ejde-761	86	19	dz	dz	PROPN
ejde-761	86	20	,	,	PUNCT
ejde-761	86	21	h	h	NOUN
ejde-761	86	22	∈	∈	NOUN
ejde-761	86	23	hθ(rn	hθ(rn	PROPN
ejde-761	86	24	)	)	PUNCT
ejde-761	87	1	⊂	⊂	PROPN
ejde-761	87	2	l2	l2	VERB
ejde-761	87	3	θ(rn	θ(rn	PROPN
ejde-761	87	4	)	)	PUNCT
ejde-761	87	5	⊂	⊂	PROPN
ejde-761	87	6	l2(rn	l2(rn	PROPN
ejde-761	87	7	)	)	PUNCT
ejde-761	87	8	.	.	PUNCT
ejde-761	88	1	the	the	DET
ejde-761	88	2	weight	weight	NOUN
ejde-761	88	3	θ(z	θ(z	NOUN
ejde-761	88	4	)	)	PUNCT
ejde-761	88	5	is	be	AUX
ejde-761	88	6	defined	define	VERB
ejde-761	88	7	in	in	ADP
ejde-761	88	8	[	[	X
ejde-761	88	9	35	35	NUM
ejde-761	88	10	,	,	PUNCT
ejde-761	88	11	26	26	NUM
ejde-761	88	12	,	,	PUNCT
ejde-761	88	13	25	25	NUM
ejde-761	88	14	]	]	PUNCT
ejde-761	88	15	as	as	ADP
ejde-761	88	16	θ(z	θ(z	NOUN
ejde-761	88	17	)	)	PUNCT
ejde-761	88	18	=	=	PUNCT
ejde-761	88	19	ec0|z|	ec0|z|	NOUN
ejde-761	88	20	4/3	4/3	NUM
ejde-761	88	21	,	,	PUNCT
ejde-761	88	22	(	(	PUNCT
ejde-761	88	23	2.4	2.4	NUM
ejde-761	88	24	)	)	PUNCT
ejde-761	88	25	where	where	SCONJ
ejde-761	88	26	c0	c0	PROPN
ejde-761	88	27	>	>	X
ejde-761	89	1	0	0	X
ejde-761	89	2	.	.	PUNCT
ejde-761	89	3	definition	definition	NOUN
ejde-761	89	4	2.4	2.4	NUM
ejde-761	89	5	.	.	PUNCT
ejde-761	90	1	the	the	DET
ejde-761	90	2	following	follow	VERB
ejde-761	90	3	fundamental	fundamental	ADJ
ejde-761	90	4	problem	problem	NOUN
ejde-761	90	5	is	be	AUX
ejde-761	90	6	defined	define	VERB
ejde-761	90	7	as	as	ADP
ejde-761	90	8	ht	ht	PROPN
ejde-761	90	9	=	=	PUNCT
ejde-761	90	10	∆m,2h	∆m,2h	NOUN
ejde-761	90	11	,	,	PUNCT
ejde-761	90	12	(	(	PUNCT
ejde-761	90	13	2.5	2.5	NUM
ejde-761	90	14	)	)	PUNCT
ejde-761	90	15	where	where	SCONJ
ejde-761	90	16	∆m,2	∆m,2	VERB
ejde-761	90	17	=	=	SYM
ejde-761	90	18	−∆(|∆	−∆(|∆	NOUN
ejde-761	90	19	·	·	PUNCT
ejde-761	90	20	|m∆	|m∆	NOUN
ejde-761	90	21	)	)	PUNCT
ejde-761	90	22	.	.	PUNCT
ejde-761	91	1	the	the	DET
ejde-761	91	2	next	next	ADJ
ejde-761	91	3	definition	definition	NOUN
ejde-761	91	4	provides	provide	VERB
ejde-761	91	5	the	the	DET
ejde-761	91	6	mollifying	mollifying	ADJ
ejde-761	91	7	exponential	exponential	ADJ
ejde-761	91	8	kernel	kernel	NOUN
ejde-761	91	9	.	.	PUNCT
ejde-761	92	1	definition	definition	NOUN
ejde-761	92	2	2.5	2.5	NUM
ejde-761	92	3	.	.	PUNCT
ejde-761	93	1	consider	consider	VERB
ejde-761	93	2	the	the	DET
ejde-761	93	3	weighted	weight	VERB
ejde-761	93	4	sobolev	sobolev	NOUN
ejde-761	93	5	norm	norm	NOUN
ejde-761	93	6	defined	define	VERB
ejde-761	93	7	as	as	ADP
ejde-761	93	8	∥h∥2hm	∥h∥2hm	NOUN
ejde-761	93	9	ρ	ρ	NOUN
ejde-761	93	10	=	=	SYM
ejde-761	93	11	∫	∫	PROPN
ejde-761	93	12	∞	∞	PROPN
ejde-761	93	13	−∞	−∞	ADP
ejde-761	93	14	emξ2	emξ2	PROPN
ejde-761	93	15	|θ(ξ	|θ(ξ	PROPN
ejde-761	93	16	,	,	PUNCT
ejde-761	93	17	t)|2dξ	t)|2dξ	INTJ
ejde-761	93	18	,	,	PUNCT
ejde-761	93	19	(	(	PUNCT
ejde-761	93	20	2.6	2.6	NUM
ejde-761	93	21	)	)	PUNCT
ejde-761	93	22	that	that	PRON
ejde-761	93	23	satisfies	satisfy	VERB
ejde-761	93	24	the	the	DET
ejde-761	93	25	ap	ap	NOUN
ejde-761	93	26	-	-	PUNCT
ejde-761	93	27	condition	condition	NOUN
ejde-761	93	28	of	of	ADP
ejde-761	93	29	mollifying	mollify	VERB
ejde-761	93	30	kernels	kernel	NOUN
ejde-761	93	31	for	for	ADP
ejde-761	93	32	p	p	NOUN
ejde-761	93	33	=	=	SYM
ejde-761	93	34	1	1	NUM
ejde-761	93	35	(	(	PUNCT
ejde-761	93	36	refer	refer	VERB
ejde-761	93	37	to	to	ADP
ejde-761	93	38	[	[	X
ejde-761	93	39	27	27	NUM
ejde-761	93	40	]	]	NUM
ejde-761	93	41	)	)	PUNCT
ejde-761	93	42	.	.	PUNCT
ejde-761	94	1	finally	finally	ADV
ejde-761	94	2	,	,	PUNCT
ejde-761	94	3	if	if	SCONJ
ejde-761	94	4	the	the	DET
ejde-761	94	5	solutions	solution	NOUN
ejde-761	94	6	are	be	AUX
ejde-761	94	7	sufficiently	sufficiently	ADV
ejde-761	94	8	far	far	ADV
ejde-761	94	9	from	from	ADP
ejde-761	94	10	the	the	DET
ejde-761	94	11	critical	critical	ADJ
ejde-761	94	12	null	null	ADJ
ejde-761	94	13	condition	condition	NOUN
ejde-761	94	14	and	and	CCONJ
ejde-761	94	15	are	be	AUX
ejde-761	94	16	sufficiently	sufficiently	ADV
ejde-761	94	17	smooth	smooth	ADJ
ejde-761	94	18	,	,	PUNCT
ejde-761	94	19	eliminating	eliminate	VERB
ejde-761	94	20	the	the	DET
ejde-761	94	21	need	need	NOUN
ejde-761	94	22	for	for	ADP
ejde-761	94	23	the	the	DET
ejde-761	94	24	previously	previously	ADV
ejde-761	94	25	defined	define	VERB
ejde-761	94	26	mollifier	mollifier	NOUN
ejde-761	94	27	,	,	PUNCT
ejde-761	94	28	we	we	PRON
ejde-761	94	29	introduce	introduce	VERB
ejde-761	94	30	the	the	DET
ejde-761	94	31	classical	classical	ADJ
ejde-761	94	32	sobolev	sobolev	NOUN
ejde-761	94	33	norm	norm	NOUN
ejde-761	94	34	.	.	PUNCT
ejde-761	95	1	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	95	2	instability	instability	NOUN
ejde-761	95	3	of	of	ADP
ejde-761	95	4	energy	energy	NOUN
ejde-761	95	5	solutions	solution	NOUN
ejde-761	95	6	5	5	NUM
ejde-761	95	7	definition	definition	NOUN
ejde-761	95	8	2.6	2.6	NUM
ejde-761	95	9	.	.	PUNCT
ejde-761	96	1	the	the	DET
ejde-761	96	2	usual	usual	ADJ
ejde-761	96	3	sobolev	sobolev	NOUN
ejde-761	96	4	order	order	NOUN
ejde-761	96	5	n	n	DET
ejde-761	96	6	functional	functional	ADJ
ejde-761	96	7	space	space	NOUN
ejde-761	96	8	is	be	AUX
ejde-761	96	9	defined	define	VERB
ejde-761	96	10	as	as	ADP
ejde-761	96	11	n(rn	n(rn	PROPN
ejde-761	96	12	)	)	PUNCT
ejde-761	97	1	=	=	PRON
ejde-761	97	2	{	{	PUNCT
ejde-761	97	3	h	h	NOUN
ejde-761	97	4	∈	∈	PROPN
ejde-761	97	5	l2(rn	l2(rn	PROPN
ejde-761	97	6	)	)	PUNCT
ejde-761	97	7	:	:	PUNCT
ejde-761	97	8	∇n(h	∇n(h	X
ejde-761	97	9	)	)	PUNCT
ejde-761	97	10	∈	∈	PROPN
ejde-761	97	11	l2(rn	l2(rn	PROPN
ejde-761	97	12	)	)	PUNCT
ejde-761	97	13	}	}	PUNCT
ejde-761	97	14	with	with	ADP
ejde-761	97	15	the	the	DET
ejde-761	97	16	norm	norm	NOUN
ejde-761	97	17	∥h∥n	∥h∥n	NOUN
ejde-761	97	18	=	=	SYM
ejde-761	97	19	∥h∥l2	∥h∥l2	NOUN
ejde-761	97	20	+	+	CCONJ
ejde-761	97	21	∥∇nh∥l2	∥∇nh∥l2	NOUN
ejde-761	97	22	.	.	PUNCT
ejde-761	98	1	(	(	PUNCT
ejde-761	98	2	2.7	2.7	NUM
ejde-761	98	3	)	)	PUNCT
ejde-761	98	4	it	it	PRON
ejde-761	98	5	should	should	AUX
ejde-761	98	6	be	be	AUX
ejde-761	98	7	mentioned	mention	VERB
ejde-761	98	8	that	that	SCONJ
ejde-761	98	9	the	the	DET
ejde-761	98	10	above	above	ADJ
ejde-761	98	11	definition	definition	NOUN
ejde-761	98	12	applies	apply	VERB
ejde-761	98	13	as	as	ADV
ejde-761	98	14	well	well	ADV
ejde-761	98	15	to	to	PART
ejde-761	98	16	compactly	compactly	ADV
ejde-761	98	17	supported	support	VERB
ejde-761	98	18	functions	function	NOUN
ejde-761	98	19	in	in	ADP
ejde-761	98	20	accordance	accordance	NOUN
ejde-761	98	21	with	with	ADP
ejde-761	98	22	the	the	DET
ejde-761	98	23	similar	similar	ADJ
ejde-761	98	24	definition	definition	NOUN
ejde-761	98	25	of	of	ADP
ejde-761	98	26	the	the	DET
ejde-761	98	27	space	space	NOUN
ejde-761	98	28	hn	hn	PROPN
ejde-761	98	29	0	0	NUM
ejde-761	98	30	(	(	PUNCT
ejde-761	98	31	rn	rn	PROPN
ejde-761	98	32	)	)	PUNCT
ejde-761	98	33	.	.	PUNCT
ejde-761	99	1	consider	consider	VERB
ejde-761	99	2	,	,	PUNCT
ejde-761	99	3	now	now	ADV
ejde-761	99	4	,	,	PUNCT
ejde-761	99	5	a	a	DET
ejde-761	99	6	sequence	sequence	NOUN
ejde-761	99	7	of	of	ADP
ejde-761	99	8	open	open	ADJ
ejde-761	99	9	bounded	bound	VERB
ejde-761	99	10	domains	domain	NOUN
ejde-761	99	11	b(0	b(0	PROPN
ejde-761	99	12	,	,	PUNCT
ejde-761	99	13	β	β	NOUN
ejde-761	99	14	)	)	PUNCT
ejde-761	100	1	⊂	⊂	PROPN
ejde-761	100	2	rn	rn	PROPN
ejde-761	100	3	,	,	PUNCT
ejde-761	100	4	β	β	PROPN
ejde-761	100	5	∈	∈	PROPN
ejde-761	100	6	n	n	CCONJ
ejde-761	100	7	,	,	PUNCT
ejde-761	100	8	β	β	X
ejde-761	100	9	=	=	SYM
ejde-761	100	10	1	1	NUM
ejde-761	100	11	,	,	PUNCT
ejde-761	100	12	2	2	NUM
ejde-761	100	13	,	,	PUNCT
ejde-761	100	14	3	3	NUM
ejde-761	100	15	.	.	PUNCT
ejde-761	100	16	.	.	PUNCT
ejde-761	100	17	.	.	PUNCT
ejde-761	101	1	proposition	proposition	NOUN
ejde-761	101	2	2.7	2.7	NUM
ejde-761	101	3	.	.	PUNCT
ejde-761	102	1	given	give	VERB
ejde-761	102	2	the	the	DET
ejde-761	102	3	sobolev	sobolev	NOUN
ejde-761	102	4	space	space	PROPN
ejde-761	102	5	wn	wn	PROPN
ejde-761	102	6	,	,	PUNCT
ejde-761	102	7	p(b(0	p(b(0	NOUN
ejde-761	102	8	,	,	PUNCT
ejde-761	102	9	β	β	NOUN
ejde-761	102	10	)	)	PUNCT
ejde-761	102	11	)	)	PUNCT
ejde-761	102	12	,	,	PUNCT
ejde-761	102	13	define	define	VERB
ejde-761	102	14	l	l	NOUN
ejde-761	102	15	=	=	SYM
ejde-761	102	16	int{n−	int{n−	PROPN
ejde-761	102	17	n	n	CCONJ
ejde-761	102	18	p	p	NOUN
ejde-761	102	19	}	}	PUNCT
ejde-761	102	20	.	.	PUNCT
ejde-761	103	1	the	the	DET
ejde-761	103	2	following	follow	VERB
ejde-761	103	3	inclusion	inclusion	NOUN
ejde-761	103	4	is	be	AUX
ejde-761	103	5	continuous	continuous	ADJ
ejde-761	103	6	(	(	PUNCT
ejde-761	103	7	see	see	VERB
ejde-761	103	8	[	[	X
ejde-761	103	9	31	31	NUM
ejde-761	103	10	,	,	PUNCT
ejde-761	103	11	p.	p.	NOUN
ejde-761	103	12	79	79	NUM
ejde-761	103	13	]	]	PUNCT
ejde-761	103	14	)	)	PUNCT
ejde-761	103	15	,	,	PUNCT
ejde-761	103	16	wn	wn	PROPN
ejde-761	103	17	,	,	PUNCT
ejde-761	103	18	p(b(0	p(b(0	NOUN
ejde-761	103	19	,	,	PUNCT
ejde-761	103	20	β	β	NOUN
ejde-761	103	21	)	)	PUNCT
ejde-761	103	22	)	)	PUNCT
ejde-761	104	1	↪	↪	PROPN
ejde-761	104	2	→	→	SYM
ejde-761	104	3	cl(b(0	cl(b(0	NOUN
ejde-761	104	4	,	,	PUNCT
ejde-761	104	5	β	β	NOUN
ejde-761	104	6	)	)	PUNCT
ejde-761	104	7	)	)	PUNCT
ejde-761	104	8	.	.	PUNCT
ejde-761	105	1	(	(	PUNCT
ejde-761	105	2	2.8	2.8	NUM
ejde-761	105	3	)	)	PUNCT
ejde-761	105	4	given	give	VERB
ejde-761	105	5	the	the	DET
ejde-761	105	6	particular	particular	ADJ
ejde-761	105	7	problem	problem	NOUN
ejde-761	105	8	in	in	ADP
ejde-761	105	9	(	(	PUNCT
ejde-761	105	10	1.1	1.1	NUM
ejde-761	105	11	)	)	PUNCT
ejde-761	105	12	,	,	PUNCT
ejde-761	105	13	any	any	DET
ejde-761	105	14	solution	solution	NOUN
ejde-761	105	15	is	be	AUX
ejde-761	105	16	,	,	PUNCT
ejde-761	105	17	at	at	ADP
ejde-761	105	18	least	least	ADJ
ejde-761	105	19	,	,	PUNCT
ejde-761	105	20	weakly	weakly	ADV
ejde-761	105	21	differentiable	differentiable	VERB
ejde-761	105	22	up	up	ADP
ejde-761	105	23	to	to	PART
ejde-761	105	24	order	order	VERB
ejde-761	105	25	four	four	NUM
ejde-761	105	26	,	,	PUNCT
ejde-761	105	27	then	then	ADV
ejde-761	105	28	n	n	NOUN
ejde-761	105	29	=	=	SYM
ejde-761	105	30	4	4	NUM
ejde-761	105	31	along	along	ADV
ejde-761	105	32	with	with	ADP
ejde-761	105	33	p	p	NOUN
ejde-761	105	34	=	=	SYM
ejde-761	105	35	2	2	NUM
ejde-761	105	36	,	,	PUNCT
ejde-761	105	37	leading	lead	VERB
ejde-761	105	38	to	to	ADP
ejde-761	105	39	l	l	NOUN
ejde-761	105	40	=	=	PUNCT
ejde-761	106	1	int{4−	int{4−	PROPN
ejde-761	106	2	d	d	NOUN
ejde-761	106	3	2	2	NUM
ejde-761	106	4	}	}	PUNCT
ejde-761	106	5	.	.	PUNCT
ejde-761	107	1	3	3	X
ejde-761	107	2	.	.	X
ejde-761	107	3	boundedness	boundedness	NOUN
ejde-761	107	4	of	of	ADP
ejde-761	107	5	solutions	solution	NOUN
ejde-761	107	6	the	the	DET
ejde-761	107	7	following	follow	VERB
ejde-761	107	8	lemma	lemma	PROPN
ejde-761	107	9	provides	provide	VERB
ejde-761	107	10	a	a	DET
ejde-761	107	11	-	-	PUNCT
ejde-761	107	12	priori	priori	ADJ
ejde-761	107	13	bounds	bound	NOUN
ejde-761	107	14	.	.	PUNCT
ejde-761	108	1	lemma	lemma	PROPN
ejde-761	108	2	3.1	3.1	NUM
ejde-761	108	3	.	.	PUNCT
ejde-761	109	1	lett	lett	PROPN
ejde-761	109	2	h	h	PROPN
ejde-761	109	3	be	be	VERB
ejde-761	109	4	a	a	DET
ejde-761	109	5	solution	solution	NOUN
ejde-761	109	6	to	to	ADP
ejde-761	109	7	the	the	DET
ejde-761	109	8	fundamental	fundamental	ADJ
ejde-761	109	9	equation	equation	NOUN
ejde-761	109	10	ht	ht	X
ejde-761	109	11	=	=	PUNCT
ejde-761	109	12	∆m,2h	∆m,2h	NOUN
ejde-761	109	13	.	.	PUNCT
ejde-761	110	1	given	give	VERB
ejde-761	110	2	h0	h0	NOUN
ejde-761	110	3	∈	∈	PROPN
ejde-761	110	4	l2(rn	l2(rn	PROPN
ejde-761	110	5	)	)	PUNCT
ejde-761	110	6	,	,	PUNCT
ejde-761	110	7	and	and	CCONJ
ejde-761	110	8	assuming	assume	VERB
ejde-761	110	9	that	that	SCONJ
ejde-761	110	10	m	m	PROPN
ejde-761	110	11	∈	∈	NOUN
ejde-761	110	12	2n	2n	NUM
ejde-761	110	13	,	,	PUNCT
ejde-761	110	14	then	then	ADV
ejde-761	110	15	the	the	DET
ejde-761	110	16	following	follow	VERB
ejde-761	110	17	bounds	bound	NOUN
ejde-761	110	18	hold	hold	VERB
ejde-761	110	19	:	:	PUNCT
ejde-761	110	20	∥h∥l2	∥h∥l2	ADJ
ejde-761	110	21	≤	≤	NUM
ejde-761	110	22	∥h0∥l2	∥h0∥l2	PUNCT
ejde-761	110	23	,	,	PUNCT
ejde-761	110	24	∥h∥hn	∥h∥hn	PROPN
ejde-761	110	25	0	0	NUM
ejde-761	110	26	≤	≤	NUM
ejde-761	110	27	∥h0∥hn	∥h0∥hn	ADP
ejde-761	110	28	0	0	NUM
ejde-761	110	29	,	,	PUNCT
ejde-761	110	30	∥h∥hm	∥h∥hm	PROPN
ejde-761	110	31	ρ	ρ	PROPN
ejde-761	110	32	≤	≤	PROPN
ejde-761	110	33	a0∥h0∥l2	a0∥h0∥l2	NUM
ejde-761	110	34	,	,	PUNCT
ejde-761	110	35	a2	a2	PROPN
ejde-761	110	36	0	0	PUNCT
ejde-761	110	37	=	=	SYM
ejde-761	110	38	e	e	X
ejde-761	110	39	mξ2m−tξ4	mξ2m−tξ4	PROPN
ejde-761	110	40	m	m	PROPN
ejde-761	110	41	γ(m+1	γ(m+1	NOUN
ejde-761	110	42	)	)	PUNCT
ejde-761	110	43	π(i	π(i	NOUN
ejde-761	110	44	ξm	ξm	AUX
ejde-761	110	45	)	)	PUNCT
ejde-761	110	46	m+1	m+1	PROPN
ejde-761	110	47	,	,	PUNCT
ejde-761	110	48	ξm	ξm	X
ejde-761	110	49	=	=	SYM
ejde-761	110	50	(	(	PUNCT
ejde-761	110	51	2mπim+1	2mπim+1	NUM
ejde-761	110	52	tγ(m+	tγ(m+	ADP
ejde-761	110	53	1)(3−m	1)(3−m	NUM
ejde-761	110	54	)	)	PUNCT
ejde-761	110	55	)	)	PUNCT
ejde-761	110	56	1	1	NUM
ejde-761	110	57	1−m	1−m	NUM
ejde-761	110	58	,	,	PUNCT
ejde-761	110	59	∥h∥hm	∥h∥hm	PROPN
ejde-761	110	60	ρ	ρ	PROPN
ejde-761	110	61	≤	≤	NUM
ejde-761	110	62	∥h0∥hm	∥h0∥hm	PUNCT
ejde-761	110	63	ρ	ρ	PROPN
ejde-761	110	64	,	,	PUNCT
ejde-761	110	65	∥h∥hm	∥h∥hm	PROPN
ejde-761	110	66	ρ	ρ	PROPN
ejde-761	110	67	≤	≤	PROPN
ejde-761	110	68	a0∥h0∥hn	a0∥h0∥hn	NOUN
ejde-761	110	69	0	0	NUM
ejde-761	110	70	,	,	PUNCT
ejde-761	110	71	∥h∥θ	∥h∥θ	ADJ
ejde-761	110	72	≤	≤	ADJ
ejde-761	110	73	σ∥h0∥hm	σ∥h0∥hm	NOUN
ejde-761	110	74	ρ	ρ	PROPN
ejde-761	110	75	,	,	PUNCT
ejde-761	110	76	∥h∥θ	∥h∥θ	ADJ
ejde-761	110	77	≤	≤	NUM
ejde-761	110	78	(	(	PUNCT
ejde-761	110	79	sup	sup	NOUN
ejde-761	110	80	z∈bβ	z∈bβ	NOUN
ejde-761	110	81	{	{	PUNCT
ejde-761	110	82	ec0|z|	ec0|z|	NOUN
ejde-761	110	83	4/3	4/3	NUM
ejde-761	110	84	}	}	PUNCT
ejde-761	110	85	)	)	PUNCT
ejde-761	110	86	1/2σ∥h0∥hn	1/2σ∥h0∥hn	NUM
ejde-761	110	87	0	0	NUM
ejde-761	110	88	,	,	PUNCT
ejde-761	110	89	where	where	SCONJ
ejde-761	110	90	σ2	σ2	NOUN
ejde-761	110	91	=	=	NOUN
ejde-761	110	92	25	25	NUM
ejde-761	110	93	sup	sup	NOUN
ejde-761	110	94	x∈bβ	x∈bβ	PROPN
ejde-761	110	95	{	{	PUNCT
ejde-761	110	96	h	h	NOUN
ejde-761	110	97	,	,	PUNCT
ejde-761	110	98	d1h	d1h	ADJ
ejde-761	110	99	,	,	PUNCT
ejde-761	110	100	d2h	d2h	PROPN
ejde-761	110	101	,	,	PUNCT
ejde-761	110	102	dh	dh	NOUN
ejde-761	110	103	,	,	PUNCT
ejde-761	110	104	d4h	d4h	NOUN
ejde-761	110	105	}	}	PUNCT
ejde-761	110	106	and	and	CCONJ
ejde-761	110	107	n	n	PRON
ejde-761	110	108	can	can	AUX
ejde-761	110	109	take	take	VERB
ejde-761	110	110	values	value	NOUN
ejde-761	110	111	1	1	NUM
ejde-761	110	112	,	,	PUNCT
ejde-761	110	113	2	2	NUM
ejde-761	110	114	,	,	PUNCT
ejde-761	110	115	3	3	NUM
ejde-761	110	116	,	,	PUNCT
ejde-761	110	117	4	4	NUM
ejde-761	110	118	.	.	X
ejde-761	110	119	note	note	VERB
ejde-761	110	120	that	that	PRON
ejde-761	110	121	bβ	bβ	NOUN
ejde-761	110	122	=	=	SYM
ejde-761	110	123	b(0	b(0	NOUN
ejde-761	110	124	,	,	PUNCT
ejde-761	110	125	β	β	NOUN
ejde-761	110	126	)	)	PUNCT
ejde-761	110	127	refers	refer	VERB
ejde-761	110	128	to	to	ADP
ejde-761	110	129	the	the	DET
ejde-761	110	130	ball	ball	NOUN
ejde-761	110	131	used	use	VERB
ejde-761	110	132	in	in	ADP
ejde-761	110	133	the	the	DET
ejde-761	110	134	proposition	proposition	NOUN
ejde-761	110	135	2	2	NUM
ejde-761	110	136	that	that	PRON
ejde-761	110	137	is	be	AUX
ejde-761	110	138	considered	consider	VERB
ejde-761	110	139	for	for	ADP
ejde-761	110	140	β	β	PROPN
ejde-761	110	141	≫	≫	PROPN
ejde-761	110	142	1	1	NUM
ejde-761	110	143	and	and	CCONJ
ejde-761	110	144	numerically	numerically	ADV
ejde-761	110	145	ordered	order	VERB
ejde-761	110	146	.	.	PUNCT
ejde-761	111	1	these	these	DET
ejde-761	111	2	last	last	ADJ
ejde-761	111	3	results	result	NOUN
ejde-761	111	4	state	state	NOUN
ejde-761	111	5	that	that	SCONJ
ejde-761	111	6	the	the	DET
ejde-761	111	7	scaling	scale	VERB
ejde-761	111	8	norm	norm	NOUN
ejde-761	111	9	hθ	hθ	VERB
ejde-761	111	10	is	be	AUX
ejde-761	111	11	bounded	bound	VERB
ejde-761	111	12	by	by	ADP
ejde-761	111	13	the	the	DET
ejde-761	111	14	mollifier	mollifier	NOUN
ejde-761	111	15	norm	norm	VERB
ejde-761	112	1	hm	hm	INTJ
ejde-761	112	2	ρ	ρ	PROPN
ejde-761	112	3	and	and	CCONJ
ejde-761	112	4	the	the	DET
ejde-761	112	5	compacting	compacting	NOUN
ejde-761	112	6	norm	norm	NOUN
ejde-761	112	7	hn	hn	PROPN
ejde-761	112	8	0	0	NUM
ejde-761	112	9	.	.	PUNCT
ejde-761	113	1	proof	proof	NOUN
ejde-761	113	2	.	.	PUNCT
ejde-761	114	1	departing	depart	VERB
ejde-761	114	2	from	from	ADP
ejde-761	114	3	the	the	DET
ejde-761	114	4	fundamental	fundamental	ADJ
ejde-761	114	5	problem	problem	NOUN
ejde-761	114	6	ht	ht	PROPN
ejde-761	114	7	=	=	SYM
ejde-761	114	8	∆m,2h	∆m,2h	NOUN
ejde-761	114	9	,	,	PUNCT
ejde-761	114	10	a	a	DET
ejde-761	114	11	general	general	ADJ
ejde-761	114	12	solution	solution	NOUN
ejde-761	114	13	is	be	AUX
ejde-761	114	14	expressed	express	VERB
ejde-761	114	15	as	as	ADP
ejde-761	114	16	h(x	h(x	PROPN
ejde-761	114	17	,	,	PUNCT
ejde-761	114	18	t	t	PROPN
ejde-761	114	19	)	)	PUNCT
ejde-761	114	20	=	=	SYM
ejde-761	114	21	et∆m,2h0(x	et∆m,2h0(x	NOUN
ejde-761	114	22	)	)	PUNCT
ejde-761	114	23	.	.	PUNCT
ejde-761	115	1	the	the	DET
ejde-761	115	2	fourier	fourier	ADJ
ejde-761	115	3	transformation	transformation	NOUN
ejde-761	115	4	(	(	PUNCT
ejde-761	115	5	in	in	ADP
ejde-761	115	6	the	the	DET
ejde-761	115	7	ξ	ξ	PROPN
ejde-761	115	8	variable	variable	NOUN
ejde-761	115	9	)	)	PUNCT
ejde-761	115	10	admits	admit	VERB
ejde-761	115	11	the	the	DET
ejde-761	115	12	following	follow	VERB
ejde-761	115	13	inequality	inequality	NOUN
ejde-761	115	14	obtained	obtain	VERB
ejde-761	115	15	by	by	ADP
ejde-761	115	16	a	a	DET
ejde-761	115	17	direct	direct	ADJ
ejde-761	115	18	convolution	convolution	NOUN
ejde-761	115	19	of	of	ADP
ejde-761	115	20	the	the	DET
ejde-761	115	21	fourier	fourier	ADJ
ejde-761	115	22	transformation	transformation	NOUN
ejde-761	115	23	for	for	ADP
ejde-761	115	24	each	each	DET
ejde-761	115	25	term	term	NOUN
ejde-761	115	26	in	in	ADP
ejde-761	115	27	(	(	PUNCT
ejde-761	115	28	|∆	|∆	NOUN
ejde-761	115	29	·	·	PUNCT
ejde-761	115	30	|m	|m	NOUN
ejde-761	115	31	∆	∆	X
ejde-761	115	32	)	)	PUNCT
ejde-761	115	33	and	and	CCONJ
ejde-761	115	34	weighted	weight	VERB
ejde-761	115	35	by	by	ADP
ejde-761	115	36	the	the	DET
ejde-761	115	37	−∆	−∆	NOUN
ejde-761	115	38	term	term	NOUN
ejde-761	115	39	,	,	PUNCT
ejde-761	115	40	ĥ(ξ	ĥ(ξ	ADJ
ejde-761	115	41	,	,	PUNCT
ejde-761	115	42	t	t	NOUN
ejde-761	115	43	)	)	PUNCT
ejde-761	115	44	≤	≤	NUM
ejde-761	116	1	e	e	PROPN
ejde-761	116	2	−tξ4	−tξ4	NOUN
ejde-761	116	3	γ(m+1	γ(m+1	PUNCT
ejde-761	116	4	)	)	PUNCT
ejde-761	116	5	2π(i	2π(i	NUM
ejde-761	117	1	ξ)m+1	ξ)m+1	ADV
ejde-761	117	2	ĥ0(ξ	ĥ0(ξ	NOUN
ejde-761	117	3	)	)	PUNCT
ejde-761	117	4	,	,	PUNCT
ejde-761	117	5	(	(	PUNCT
ejde-761	117	6	3.1	3.1	NUM
ejde-761	117	7	)	)	PUNCT
ejde-761	117	8	where	where	SCONJ
ejde-761	117	9	γ(m+	γ(m+	NUM
ejde-761	117	10	1	1	NUM
ejde-761	117	11	)	)	PUNCT
ejde-761	117	12	refers	refer	VERB
ejde-761	117	13	to	to	ADP
ejde-761	117	14	the	the	DET
ejde-761	117	15	gamma	gamma	NOUN
ejde-761	117	16	function	function	NOUN
ejde-761	117	17	and	and	CCONJ
ejde-761	117	18	i	i	PRON
ejde-761	117	19	to	to	ADP
ejde-761	117	20	the	the	DET
ejde-761	117	21	imaginary	imaginary	ADJ
ejde-761	117	22	unit	unit	NOUN
ejde-761	117	23	.	.	PUNCT
ejde-761	118	1	firstly	firstly	ADV
ejde-761	118	2	,	,	PUNCT
ejde-761	118	3	the	the	DET
ejde-761	118	4	following	follow	VERB
ejde-761	118	5	bound	bind	VERB
ejde-761	118	6	is	be	AUX
ejde-761	118	7	shown	show	VERB
ejde-761	118	8	for	for	ADP
ejde-761	118	9	m	m	PROPN
ejde-761	118	10	∈	∈	PROPN
ejde-761	118	11	2n	2n	NUM
ejde-761	118	12	,	,	PUNCT
ejde-761	118	13	∥h∥2l2	∥h∥2l2	PROPN
ejde-761	118	14	≤	≤	NUM
ejde-761	118	15	∫	∫	PROPN
ejde-761	118	16	|e−tξ4	|e−tξ4	PROPN
ejde-761	118	17	γ(m+1	γ(m+1	NOUN
ejde-761	118	18	)	)	PUNCT
ejde-761	118	19	π(i	π(i	NOUN
ejde-761	118	20	ξ)m+1	ξ)m+1	PRON
ejde-761	119	1	|∥ĥ0(ξ)∥2dξ	|∥ĥ0(ξ)∥2dξ	PROPN
ejde-761	119	2	≤	≤	NUM
ejde-761	119	3	sup	sup	NOUN
ejde-761	119	4	∀ξ∈rn	∀ξ∈rn	NUM
ejde-761	119	5	{	{	PUNCT
ejde-761	119	6	|e−tξ4	|e−tξ4	NOUN
ejde-761	119	7	γ(m+1	γ(m+1	NUM
ejde-761	119	8	)	)	PUNCT
ejde-761	119	9	π(i	π(i	PROPN
ejde-761	119	10	ξ)m+1	ξ)m+1	ADV
ejde-761	120	1	|	|	ADV
ejde-761	120	2	}	}	PUNCT
ejde-761	120	3	∫	∫	PROPN
ejde-761	120	4	∥ĥ0(ξ)∥2dξ	∥ĥ0(ξ)∥2dξ	X
ejde-761	120	5	=	=	PUNCT
ejde-761	121	1	∥h0∥2l2	∥h0∥2l2	PROPN
ejde-761	121	2	.	.	PUNCT
ejde-761	122	1	6	6	NUM
ejde-761	122	2	j.	j.	PROPN
ejde-761	122	3	l.	l.	PROPN
ejde-761	122	4	díaz	díaz	PROPN
ejde-761	122	5	palencia	palencia	PROPN
ejde-761	122	6	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	122	7	then	then	ADV
ejde-761	122	8	∥h∥l2	∥h∥l2	VERB
ejde-761	122	9	≤	≤	NOUN
ejde-761	122	10	∥h0∥l2	∥h0∥l2	PUNCT
ejde-761	122	11	.	.	PUNCT
ejde-761	123	1	as	as	ADP
ejde-761	123	2	a	a	DET
ejde-761	123	3	direct	direct	ADJ
ejde-761	123	4	consequence	consequence	NOUN
ejde-761	123	5	,	,	PUNCT
ejde-761	123	6	and	and	CCONJ
ejde-761	123	7	considering	consider	VERB
ejde-761	123	8	the	the	DET
ejde-761	123	9	expression	expression	NOUN
ejde-761	123	10	in	in	ADP
ejde-761	123	11	(	(	PUNCT
ejde-761	123	12	2.7	2.7	NUM
ejde-761	123	13	)	)	PUNCT
ejde-761	123	14	,	,	PUNCT
ejde-761	123	15	we	we	PRON
ejde-761	123	16	have	have	VERB
ejde-761	123	17	∥h∥hn	∥h∥hn	PROPN
ejde-761	123	18	0	0	NUM
ejde-761	123	19	≤	≤	NUM
ejde-761	123	20	∥h0∥hn	∥h0∥hn	ADP
ejde-761	123	21	0	0	NUM
ejde-761	123	22	.	.	PUNCT
ejde-761	124	1	(	(	PUNCT
ejde-761	124	2	3.2	3.2	NUM
ejde-761	124	3	)	)	PUNCT
ejde-761	124	4	now	now	ADV
ejde-761	124	5	,	,	PUNCT
ejde-761	124	6	assume	assume	VERB
ejde-761	124	7	that	that	SCONJ
ejde-761	124	8	h0	h0	NOUN
ejde-761	124	9	∈	∈	PROPN
ejde-761	124	10	l2(rn	l2(rn	PROPN
ejde-761	124	11	)	)	PUNCT
ejde-761	124	12	.	.	PUNCT
ejde-761	125	1	then	then	ADV
ejde-761	125	2	∥h∥2hm	∥h∥2hm	VERB
ejde-761	125	3	ρ	ρ	PROPN
ejde-761	125	4	=	=	SYM
ejde-761	125	5	∫	∫	PROPN
ejde-761	125	6	∞	∞	PROPN
ejde-761	125	7	−∞	−∞	ADP
ejde-761	125	8	emξ2	emξ2	NOUN
ejde-761	125	9	|ĥ(ξ	|ĥ(ξ	NUM
ejde-761	125	10	,	,	PUNCT
ejde-761	125	11	t)|2dξ	t)|2dξ	NOUN
ejde-761	125	12	≤	≤	NUM
ejde-761	125	13	sup	sup	NOUN
ejde-761	125	14	∀ξ∈rn	∀ξ∈rn	NUM
ejde-761	125	15	{	{	PUNCT
ejde-761	125	16	emξ2e	emξ2e	PROPN
ejde-761	125	17	−tξ4	−tξ4	NOUN
ejde-761	125	18	γ(m+1	γ(m+1	NUM
ejde-761	125	19	)	)	PUNCT
ejde-761	125	20	π(i	π(i	NOUN
ejde-761	125	21	ξ)m+1	ξ)m+1	ADV
ejde-761	125	22	}	}	PUNCT
ejde-761	125	23	∫	∫	PROPN
ejde-761	125	24	∥ĥ0(ξ)∥2dξ	∥ĥ0(ξ)∥2dξ	PROPN
ejde-761	125	25	.	.	PUNCT
ejde-761	126	1	(	(	PUNCT
ejde-761	126	2	3.3	3.3	NUM
ejde-761	126	3	)	)	PUNCT
ejde-761	126	4	making	make	VERB
ejde-761	126	5	standard	standard	ADJ
ejde-761	126	6	operations	operation	NOUN
ejde-761	126	7	,	,	PUNCT
ejde-761	126	8	the	the	DET
ejde-761	126	9	following	follow	VERB
ejde-761	126	10	holds	hold	VERB
ejde-761	126	11	∥h∥2hm	∥h∥2hm	ADJ
ejde-761	126	12	ρ	ρ	PROPN
ejde-761	126	13	≤	≤	X
ejde-761	126	14	e	e	X
ejde-761	126	15	mξ2m−tξ4	mξ2m−tξ4	PROPN
ejde-761	126	16	m	m	PRON
ejde-761	126	17	γ(m+1	γ(m+1	NOUN
ejde-761	126	18	)	)	PUNCT
ejde-761	126	19	π(i	π(i	NOUN
ejde-761	126	20	ξm	ξm	VERB
ejde-761	126	21	)	)	PUNCT
ejde-761	126	22	m+1	m+1	NUM
ejde-761	126	23	∥h0∥2l2	∥h0∥2l2	X
ejde-761	126	24	,	,	PUNCT
ejde-761	126	25	(	(	PUNCT
ejde-761	126	26	3.4	3.4	NUM
ejde-761	126	27	)	)	PUNCT
ejde-761	126	28	where	where	SCONJ
ejde-761	126	29	ξm	ξm	VERB
ejde-761	126	30	=	=	X
ejde-761	126	31	(	(	PUNCT
ejde-761	126	32	2mπim+1	2mπim+1	NUM
ejde-761	126	33	tγ(m+	tγ(m+	ADP
ejde-761	126	34	1)(3−m	1)(3−m	NUM
ejde-761	126	35	)	)	PUNCT
ejde-761	126	36	)	)	PUNCT
ejde-761	126	37	1	1	NUM
ejde-761	126	38	1−m	1−m	NUM
ejde-761	126	39	.	.	PUNCT
ejde-761	127	1	(	(	PUNCT
ejde-761	127	2	3.5	3.5	NUM
ejde-761	127	3	)	)	PUNCT
ejde-761	127	4	since	since	SCONJ
ejde-761	127	5	h0	h0	NOUN
ejde-761	127	6	∈	∈	PROPN
ejde-761	127	7	l2(rn	l2(rn	PROPN
ejde-761	127	8	)	)	PUNCT
ejde-761	127	9	,	,	PUNCT
ejde-761	127	10	the	the	DET
ejde-761	127	11	rapid	rapid	ADJ
ejde-761	127	12	decay	decay	NOUN
ejde-761	127	13	of	of	ADP
ejde-761	127	14	e−tξ4·c(m	e−tξ4·c(m	NOUN
ejde-761	127	15	,	,	PUNCT
ejde-761	127	16	iξ	iξ	ADP
ejde-761	127	17	)	)	PUNCT
ejde-761	127	18	,	,	PUNCT
ejde-761	127	19	where	where	SCONJ
ejde-761	127	20	c(m	c(m	PROPN
ejde-761	127	21	,	,	PUNCT
ejde-761	127	22	iξ	iξ	NOUN
ejde-761	127	23	)	)	PUNCT
ejde-761	127	24	is	be	AUX
ejde-761	127	25	a	a	DET
ejde-761	127	26	constant	constant	ADJ
ejde-761	127	27	that	that	PRON
ejde-761	127	28	depend	depend	VERB
ejde-761	127	29	on	on	ADP
ejde-761	127	30	m	m	PROPN
ejde-761	127	31	and	and	CCONJ
ejde-761	127	32	the	the	DET
ejde-761	127	33	imaginary	imaginary	ADJ
ejde-761	127	34	unit	unit	NOUN
ejde-761	127	35	,	,	PUNCT
ejde-761	127	36	ensures	ensure	VERB
ejde-761	127	37	the	the	DET
ejde-761	127	38	integral	integral	ADJ
ejde-761	127	39	∫	∫	PROPN
ejde-761	127	40	emξ2	emξ2	PROPN
ejde-761	127	41	|ĥ0(ξ	|ĥ0(ξ	NOUN
ejde-761	127	42	,	,	PUNCT
ejde-761	127	43	t)|2dξ	t)|2dξ	PRON
ejde-761	127	44	is	be	AUX
ejde-761	127	45	finite	finite	ADJ
ejde-761	127	46	;	;	PUNCT
ejde-761	127	47	thus	thus	ADV
ejde-761	127	48	,	,	PUNCT
ejde-761	127	49	h0	h0	PROPN
ejde-761	127	50	∈	∈	PROPN
ejde-761	127	51	hm	hm	INTJ
ejde-761	127	52	ρ	ρ	PROPN
ejde-761	127	53	.	.	PUNCT
ejde-761	128	1	we	we	PRON
ejde-761	128	2	show	show	VERB
ejde-761	128	3	then	then	ADV
ejde-761	128	4	that	that	SCONJ
ejde-761	128	5	any	any	DET
ejde-761	128	6	solution	solution	NOUN
ejde-761	128	7	,	,	PUNCT
ejde-761	128	8	h	h	NOUN
ejde-761	128	9	,	,	PUNCT
ejde-761	128	10	to	to	ADP
ejde-761	128	11	the	the	DET
ejde-761	128	12	fundamental	fundamental	ADJ
ejde-761	128	13	equation	equation	NOUN
ejde-761	128	14	in	in	ADP
ejde-761	128	15	hm	hm	INTJ
ejde-761	128	16	ρ	ρ	PROPN
ejde-761	128	17	satisfies	satisfie	NOUN
ejde-761	128	18	∥h∥2hm	∥h∥2hm	VERB
ejde-761	128	19	ρ	ρ	PROPN
ejde-761	128	20	=	=	SYM
ejde-761	128	21	∫	∫	PROPN
ejde-761	128	22	emξ2	emξ2	PROPN
ejde-761	128	23	|ĥ(ξ	|ĥ(ξ	PROPN
ejde-761	128	24	,	,	PUNCT
ejde-761	128	25	t)|2dξ	t)|2dξ	NOUN
ejde-761	128	26	≤	≤	NUM
ejde-761	128	27	sup	sup	NOUN
ejde-761	128	28	∀ξ∈rn	∀ξ∈rn	NUM
ejde-761	128	29	{	{	PUNCT
ejde-761	128	30	|e−tξ4	|e−tξ4	NOUN
ejde-761	128	31	γ(m+1	γ(m+1	NUM
ejde-761	128	32	)	)	PUNCT
ejde-761	128	33	π(iξ)m+1	π(iξ)m+1	VERB
ejde-761	128	34	|	|	ADV
ejde-761	128	35	}	}	PUNCT
ejde-761	128	36	∫	∫	PROPN
ejde-761	128	37	emξ2∥ĥ0(ξ)∥2dξ	emξ2∥ĥ0(ξ)∥2dξ	PROPN
ejde-761	128	38	≤	≤	PROPN
ejde-761	128	39	∥h0∥2hm	∥h0∥2hm	PROPN
ejde-761	128	40	ρ	ρ	NOUN
ejde-761	128	41	.	.	PUNCT
ejde-761	129	1	(	(	PUNCT
ejde-761	129	2	3.6	3.6	NUM
ejde-761	129	3	)	)	PUNCT
ejde-761	129	4	the	the	DET
ejde-761	129	5	following	follow	VERB
ejde-761	129	6	bound	bind	VERB
ejde-761	129	7	is	be	AUX
ejde-761	129	8	also	also	ADV
ejde-761	129	9	applicable	applicable	ADJ
ejde-761	129	10	:	:	PUNCT
ejde-761	129	11	∥h∥2hm	∥h∥2hm	PROPN
ejde-761	129	12	ρ	ρ	PROPN
ejde-761	129	13	=	=	SYM
ejde-761	129	14	∫	∫	PROPN
ejde-761	129	15	emξ2	emξ2	PROPN
ejde-761	129	16	|ĥ(ξ	|ĥ(ξ	PROPN
ejde-761	129	17	,	,	PUNCT
ejde-761	129	18	t)|2dξ	t)|2dξ	NOUN
ejde-761	129	19	≤	≤	NUM
ejde-761	129	20	sup	sup	NOUN
ejde-761	129	21	∀ξ∈rn	∀ξ∈rn	NUM
ejde-761	129	22	{	{	PUNCT
ejde-761	129	23	|emξ2−tξ4	|emξ2−tξ4	PROPN
ejde-761	129	24	γ(m+1	γ(m+1	PUNCT
ejde-761	129	25	)	)	PUNCT
ejde-761	129	26	π(i	π(i	PROPN
ejde-761	129	27	ξ)m+1	ξ)m+1	ADV
ejde-761	130	1	|	|	ADV
ejde-761	130	2	}	}	PUNCT
ejde-761	130	3	∫	∫	PROPN
ejde-761	130	4	∥ĥ0(ξ)∥2dξ	∥ĥ0(ξ)∥2dξ	PROPN
ejde-761	130	5	≤	≤	NUM
ejde-761	130	6	sup	sup	NOUN
ejde-761	130	7	∀ξ∈rn	∀ξ∈rn	NUM
ejde-761	130	8	{	{	PUNCT
ejde-761	130	9	|emξ2−tξ4	|emξ2−tξ4	PROPN
ejde-761	130	10	γ(m+1	γ(m+1	PUNCT
ejde-761	130	11	)	)	PUNCT
ejde-761	130	12	π(i	π(i	PROPN
ejde-761	130	13	ξ)m+1	ξ)m+1	ADV
ejde-761	130	14	|	|	CCONJ
ejde-761	130	15	}	}	PUNCT
ejde-761	130	16	∫	∫	PROPN
ejde-761	130	17	(	(	PUNCT
ejde-761	130	18	∥ĥ0(ξ)∥2	∥ĥ0(ξ)∥2	PROPN
ejde-761	130	19	+	+	NUM
ejde-761	130	20	|∇nh0|2)dξ	|∇nh0|2)dξ	X
ejde-761	130	21	=	=	X
ejde-761	130	22	e	e	X
ejde-761	130	23	mξ2m−tξ4	mξ2m−tξ4	PROPN
ejde-761	130	24	m	m	PROPN
ejde-761	130	25	γ(m+1	γ(m+1	NOUN
ejde-761	130	26	)	)	PUNCT
ejde-761	130	27	π(iξm	π(iξm	NOUN
ejde-761	130	28	)	)	PUNCT
ejde-761	131	1	m+1	m+1	NUM
ejde-761	131	2	∥h0∥2hn	∥h0∥2hn	PROPN
ejde-761	131	3	0	0	NUM
ejde-761	131	4	,	,	PUNCT
ejde-761	131	5	(	(	PUNCT
ejde-761	131	6	3.7	3.7	NUM
ejde-761	131	7	)	)	PUNCT
ejde-761	131	8	where	where	SCONJ
ejde-761	131	9	ξm	ξm	PROPN
ejde-761	131	10	is	be	AUX
ejde-761	131	11	given	give	VERB
ejde-761	131	12	in	in	ADP
ejde-761	131	13	(	(	PUNCT
ejde-761	131	14	3.5	3.5	NUM
ejde-761	131	15	)	)	PUNCT
ejde-761	131	16	.	.	PUNCT
ejde-761	132	1	now	now	ADV
ejde-761	132	2	,	,	PUNCT
ejde-761	132	3	the	the	DET
ejde-761	132	4	intention	intention	NOUN
ejde-761	132	5	is	be	AUX
ejde-761	132	6	to	to	PART
ejde-761	132	7	show	show	VERB
ejde-761	132	8	a	a	DET
ejde-761	132	9	bound	bind	VERB
ejde-761	132	10	for	for	ADP
ejde-761	132	11	the	the	DET
ejde-761	132	12	norm	norm	NOUN
ejde-761	132	13	defined	define	VERB
ejde-761	132	14	in	in	ADP
ejde-761	132	15	(	(	PUNCT
ejde-761	132	16	2.3	2.3	NUM
ejde-761	132	17	)	)	PUNCT
ejde-761	132	18	.	.	PUNCT
ejde-761	133	1	the	the	DET
ejde-761	133	2	bounding	bounding	NOUN
ejde-761	133	3	term	term	NOUN
ejde-761	133	4	is	be	AUX
ejde-761	133	5	given	give	VERB
ejde-761	133	6	by	by	ADP
ejde-761	133	7	the	the	DET
ejde-761	133	8	mollifier	mollifier	ADJ
ejde-761	133	9	norm	norm	NOUN
ejde-761	133	10	introduced	introduce	VERB
ejde-761	133	11	in	in	ADP
ejde-761	133	12	(	(	PUNCT
ejde-761	133	13	2.6	2.6	NUM
ejde-761	133	14	):	):	PUNCT
ejde-761	133	15	∥h∥2θ	∥h∥2θ	PROPN
ejde-761	133	16	=	=	SYM
ejde-761	133	17	∫	∫	PROPN
ejde-761	133	18	θ(z	θ(z	NOUN
ejde-761	133	19	)	)	PUNCT
ejde-761	133	20	4∑	4∑	NOUN
ejde-761	133	21	j=0	j=0	VERB
ejde-761	133	22	|djh(z)|2dz	|djh(z)|2dz	NOUN
ejde-761	133	23	≤	≤	NUM
ejde-761	133	24	∫	∫	PROPN
ejde-761	133	25	emz2	emz2	PROPN
ejde-761	133	26	4∑	4∑	PROPN
ejde-761	133	27	j=0	j=0	VERB
ejde-761	133	28	|djh(z)|2dz	|djh(z)|2dz	NOUN
ejde-761	133	29	≤	≤	NUM
ejde-761	133	30	σ2	σ2	PROPN
ejde-761	133	31	∫	∫	PROPN
ejde-761	133	32	emz2	emz2	NOUN
ejde-761	133	33	|h(z)|2dz	|h(z)|2dz	NOUN
ejde-761	133	34	=	=	PUNCT
ejde-761	133	35	σ2∥h∥2hm	σ2∥h∥2hm	NUM
ejde-761	133	36	ρ	ρ	NUM
ejde-761	133	37	≤	≤	NUM
ejde-761	133	38	σ2∥h0∥2hm	σ2∥h0∥2hm	PROPN
ejde-761	133	39	ρ	ρ	NOUN
ejde-761	133	40	,	,	PUNCT
ejde-761	133	41	(	(	PUNCT
ejde-761	133	42	3.8	3.8	NUM
ejde-761	133	43	)	)	PUNCT
ejde-761	133	44	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	133	45	instability	instability	NOUN
ejde-761	133	46	of	of	ADP
ejde-761	133	47	energy	energy	NOUN
ejde-761	133	48	solutions	solution	NOUN
ejde-761	133	49	7	7	NUM
ejde-761	133	50	being	being	NOUN
ejde-761	133	51	σ2	σ2	NOUN
ejde-761	133	52	=	=	NOUN
ejde-761	133	53	25	25	NUM
ejde-761	133	54	supx∈bβ	supx∈bβ	NOUN
ejde-761	133	55	{	{	PUNCT
ejde-761	133	56	h	h	NOUN
ejde-761	133	57	,	,	PUNCT
ejde-761	133	58	d1h	d1h	ADJ
ejde-761	133	59	,	,	PUNCT
ejde-761	133	60	d2h	d2h	PROPN
ejde-761	133	61	,	,	PUNCT
ejde-761	133	62	dh	dh	NOUN
ejde-761	133	63	,	,	PUNCT
ejde-761	133	64	d4h	d4h	NOUN
ejde-761	133	65	}	}	PUNCT
ejde-761	133	66	,	,	PUNCT
ejde-761	133	67	for	for	ADP
ejde-761	133	68	β	β	PROPN
ejde-761	133	69	≫	≫	PROPN
ejde-761	133	70	1	1	NUM
ejde-761	133	71	and	and	CCONJ
ejde-761	133	72	ordered	order	VERB
ejde-761	133	73	.	.	PUNCT
ejde-761	134	1	the	the	DET
ejde-761	134	2	continuity	continuity	NOUN
ejde-761	134	3	inclusion	inclusion	NOUN
ejde-761	134	4	in	in	ADP
ejde-761	134	5	the	the	DET
ejde-761	134	6	proposition	proposition	NOUN
ejde-761	134	7	2.7	2.7	NUM
ejde-761	134	8	provides	provide	VERB
ejde-761	134	9	the	the	DET
ejde-761	134	10	conditions	condition	NOUN
ejde-761	134	11	to	to	PART
ejde-761	134	12	ensure	ensure	VERB
ejde-761	134	13	the	the	DET
ejde-761	134	14	existence	existence	NOUN
ejde-761	134	15	of	of	ADP
ejde-761	134	16	the	the	DET
ejde-761	134	17	derivatives	derivative	NOUN
ejde-761	134	18	of	of	ADP
ejde-761	134	19	a	a	DET
ejde-761	134	20	function	function	NOUN
ejde-761	134	21	h	h	NOUN
ejde-761	134	22	∈	∈	PROPN
ejde-761	134	23	w	w	PROPN
ejde-761	134	24	4,2(b(0	4,2(b(0	NUM
ejde-761	134	25	,	,	PUNCT
ejde-761	134	26	β	β	NOUN
ejde-761	134	27	)	)	PUNCT
ejde-761	134	28	)	)	PUNCT
ejde-761	134	29	.	.	PUNCT
ejde-761	135	1	in	in	ADP
ejde-761	135	2	addition	addition	NOUN
ejde-761	135	3	,	,	PUNCT
ejde-761	135	4	the	the	DET
ejde-761	135	5	following	follow	VERB
ejde-761	135	6	holds	hold	VERB
ejde-761	135	7	:	:	PUNCT
ejde-761	135	8	∥h∥2θ	∥h∥2θ	PROPN
ejde-761	135	9	=	=	SYM
ejde-761	135	10	∫	∫	PROPN
ejde-761	135	11	θ(z	θ(z	NOUN
ejde-761	135	12	)	)	PUNCT
ejde-761	135	13	4∑	4∑	NOUN
ejde-761	135	14	j=0	j=0	VERB
ejde-761	135	15	|djh(z)|2dz	|djh(z)|2dz	NOUN
ejde-761	135	16	≤	≤	NUM
ejde-761	135	17	sup	sup	NOUN
ejde-761	135	18	∀z∈bβ	∀z∈bβ	NUM
ejde-761	135	19	{	{	PUNCT
ejde-761	135	20	ec0|z|	ec0|z|	NOUN
ejde-761	135	21	4/3	4/3	NUM
ejde-761	135	22	}	}	PUNCT
ejde-761	135	23	σ2	σ2	PROPN
ejde-761	135	24	∫	∫	PROPN
ejde-761	135	25	(	(	PUNCT
ejde-761	135	26	|h(z)|2	|h(z)|2	X
ejde-761	135	27	+	+	PUNCT
ejde-761	135	28	|∇nh|2)dz	|∇nh|2)dz	PROPN
ejde-761	135	29	≤	≤	NUM
ejde-761	135	30	sup	sup	NOUN
ejde-761	135	31	∀z∈bβ	∀z∈bβ	NUM
ejde-761	135	32	{	{	PUNCT
ejde-761	135	33	ec0|z|	ec0|z|	NOUN
ejde-761	135	34	4/3	4/3	NUM
ejde-761	135	35	}	}	PUNCT
ejde-761	135	36	σ2∥h∥2hn	σ2∥h∥2hn	PROPN
ejde-761	135	37	0	0	NUM
ejde-761	135	38	≤	≤	NUM
ejde-761	135	39	sup	sup	NOUN
ejde-761	135	40	∀z∈bβ	∀z∈bβ	NUM
ejde-761	135	41	{	{	PUNCT
ejde-761	135	42	ec0|z|	ec0|z|	NOUN
ejde-761	135	43	4/3	4/3	PRON
ejde-761	135	44	}	}	PUNCT
ejde-761	135	45	σ2∥h0∥2hn	σ2∥h0∥2hn	NOUN
ejde-761	135	46	0	0	NUM
ejde-761	135	47	,	,	PUNCT
ejde-761	135	48	(	(	PUNCT
ejde-761	135	49	3.9	3.9	NUM
ejde-761	135	50	)	)	PUNCT
ejde-761	135	51	where	where	SCONJ
ejde-761	135	52	n	n	PRON
ejde-761	135	53	can	can	AUX
ejde-761	135	54	take	take	VERB
ejde-761	135	55	a	a	DET
ejde-761	135	56	value	value	NOUN
ejde-761	135	57	1	1	NUM
ejde-761	135	58	,	,	PUNCT
ejde-761	135	59	2	2	NUM
ejde-761	135	60	,	,	PUNCT
ejde-761	135	61	3	3	NUM
ejde-761	135	62	,	,	PUNCT
ejde-761	135	63	4	4	NUM
ejde-761	135	64	and	and	CCONJ
ejde-761	135	65	β	β	X
ejde-761	135	66	≫	≫	PROPN
ejde-761	135	67	1	1	NUM
ejde-761	135	68	and	and	CCONJ
ejde-761	135	69	ordered	order	VERB
ejde-761	135	70	to	to	PART
ejde-761	135	71	expand	expand	VERB
ejde-761	135	72	up	up	ADP
ejde-761	135	73	to	to	ADP
ejde-761	135	74	the	the	DET
ejde-761	135	75	whole	whole	ADJ
ejde-761	135	76	rn	rn	PROPN
ejde-761	135	77	.	.	PROPN
ejde-761	135	78	based	base	VERB
ejde-761	135	79	on	on	ADP
ejde-761	135	80	the	the	DET
ejde-761	135	81	proposed	propose	VERB
ejde-761	135	82	arguments	argument	NOUN
ejde-761	135	83	,	,	PUNCT
ejde-761	135	84	the	the	DET
ejde-761	135	85	lemma	lemma	PROPN
ejde-761	135	86	postulations	postulation	NOUN
ejde-761	135	87	are	be	AUX
ejde-761	135	88	proved	prove	VERB
ejde-761	135	89	.	.	PUNCT
ejde-761	136	1	□	□	PUNCT
ejde-761	136	2	the	the	DET
ejde-761	136	3	next	next	ADJ
ejde-761	136	4	objective	objective	NOUN
ejde-761	136	5	is	be	AUX
ejde-761	136	6	to	to	PART
ejde-761	136	7	show	show	VERB
ejde-761	136	8	the	the	DET
ejde-761	136	9	local	local	ADJ
ejde-761	136	10	bound	bind	VERB
ejde-761	136	11	properties	property	NOUN
ejde-761	136	12	of	of	ADP
ejde-761	136	13	compactly	compactly	ADV
ejde-761	136	14	supported	support	VERB
ejde-761	136	15	solutions	solution	NOUN
ejde-761	136	16	.	.	PUNCT
ejde-761	137	1	to	to	ADP
ejde-761	137	2	this	this	DET
ejde-761	137	3	end	end	NOUN
ejde-761	137	4	,	,	PUNCT
ejde-761	137	5	we	we	PRON
ejde-761	137	6	assume	assume	VERB
ejde-761	137	7	that	that	SCONJ
ejde-761	137	8	the	the	DET
ejde-761	137	9	following	follow	VERB
ejde-761	137	10	conditions	condition	NOUN
ejde-761	137	11	hold	hold	VERB
ejde-761	137	12	:	:	PUNCT
ejde-761	137	13	η	η	PROPN
ejde-761	137	14	∈	∈	PROPN
ejde-761	137	15	l1(0	l1(0	PROPN
ejde-761	137	16	,	,	PUNCT
ejde-761	137	17	τ	τ	PROPN
ejde-761	137	18	;	;	PUNCT
ejde-761	137	19	h	h	PROPN
ejde-761	137	20	2	2	NUM
ejde-761	137	21	0	0	NUM
ejde-761	137	22	(	(	PUNCT
ejde-761	137	23	rn	rn	PROPN
ejde-761	137	24	)	)	PUNCT
ejde-761	137	25	)	)	PUNCT
ejde-761	138	1	∩	∩	PROPN
ejde-761	138	2	c2(0	c2(0	PROPN
ejde-761	138	3	,	,	PUNCT
ejde-761	138	4	τ	τ	X
ejde-761	138	5	;	;	PUNCT
ejde-761	138	6	h2	h2	PROPN
ejde-761	138	7	0	0	NUM
ejde-761	138	8	∩	∩	PROPN
ejde-761	138	9	c	c	PROPN
ejde-761	138	10	n	n	X
ejde-761	138	11	0	0	NUM
ejde-761	138	12	(	(	PUNCT
ejde-761	138	13	rn	rn	PROPN
ejde-761	138	14	)	)	PUNCT
ejde-761	138	15	)	)	PUNCT
ejde-761	138	16	,	,	PUNCT
ejde-761	138	17	u0(x	u0(x	NOUN
ejde-761	138	18	)	)	PUNCT
ejde-761	138	19	∈	∈	NOUN
ejde-761	138	20	hn	hn	INTJ
ejde-761	138	21	0	0	NUM
ejde-761	138	22	(	(	PUNCT
ejde-761	138	23	rn	rn	PROPN
ejde-761	138	24	)	)	PUNCT
ejde-761	138	25	∩	∩	PROPN
ejde-761	138	26	cn	cn	X
ejde-761	138	27	0	0	SYM
ejde-761	138	28	(	(	PUNCT
ejde-761	138	29	rn	rn	PROPN
ejde-761	138	30	)	)	PUNCT
ejde-761	138	31	,	,	PUNCT
ejde-761	138	32	(	(	PUNCT
ejde-761	138	33	3.10	3.10	NUM
ejde-761	138	34	)	)	PUNCT
ejde-761	138	35	with	with	ADP
ejde-761	138	36	n	n	PRON
ejde-761	138	37	≥	≥	NUM
ejde-761	138	38	1	1	NUM
ejde-761	138	39	.	.	PUNCT
ejde-761	139	1	lemma	lemma	PROPN
ejde-761	139	2	3.2	3.2	NUM
ejde-761	139	3	.	.	PUNCT
ejde-761	140	1	each	each	DET
ejde-761	140	2	energy	energy	NOUN
ejde-761	140	3	solution	solution	NOUN
ejde-761	140	4	satisfying	satisfying	ADJ
ejde-761	140	5	(	(	PUNCT
ejde-761	140	6	2.1	2.1	NUM
ejde-761	140	7	)	)	PUNCT
ejde-761	140	8	is	be	AUX
ejde-761	140	9	bounded	bound	VERB
ejde-761	140	10	in	in	ADP
ejde-761	140	11	h4	h4	PROPN
ejde-761	140	12	0	0	NUM
ejde-761	140	13	(	(	PUNCT
ejde-761	140	14	rn	rn	PROPN
ejde-761	140	15	)	)	PUNCT
ejde-761	140	16	(	(	PUNCT
ejde-761	140	17	see	see	VERB
ejde-761	140	18	norm	norm	NOUN
ejde-761	140	19	(	(	PUNCT
ejde-761	140	20	2.7	2.7	NUM
ejde-761	140	21	)	)	PUNCT
ejde-761	140	22	)	)	PUNCT
ejde-761	140	23	,	,	PUNCT
ejde-761	140	24	i.e.	i.e.	X
ejde-761	140	25	the	the	DET
ejde-761	140	26	compact	compact	ADJ
ejde-761	140	27	support	support	NOUN
ejde-761	140	28	is	be	AUX
ejde-761	140	29	locally	locally	ADV
ejde-761	140	30	preserved	preserve	VERB
ejde-761	140	31	.	.	PUNCT
ejde-761	141	1	proof	proof	NOUN
ejde-761	141	2	.	.	PUNCT
ejde-761	142	1	firstly	firstly	ADV
ejde-761	142	2	,	,	PUNCT
ejde-761	142	3	the	the	DET
ejde-761	142	4	expression	expression	NOUN
ejde-761	142	5	(	(	PUNCT
ejde-761	142	6	2.1	2.1	NUM
ejde-761	142	7	)	)	PUNCT
ejde-761	142	8	is	be	AUX
ejde-761	142	9	rewritten	rewrite	VERB
ejde-761	142	10	as∫	as∫	PROPN
ejde-761	143	1	τ	τ	PROPN
ejde-761	143	2	0	0	NUM
ejde-761	143	3	∫	∫	PROPN
ejde-761	143	4	rn	rn	PROPN
ejde-761	143	5	utη	utη	PROPN
ejde-761	143	6	dx	dx	PROPN
ejde-761	143	7	dt+	dt+	NOUN
ejde-761	143	8	∫	∫	PROPN
ejde-761	143	9	τ	τ	X
ejde-761	143	10	0	0	NUM
ejde-761	143	11	∫	∫	PROPN
ejde-761	143	12	rn	rn	PROPN
ejde-761	143	13	|u|p−1uη	|u|p−1uη	PROPN
ejde-761	143	14	dx	dx	PROPN
ejde-761	143	15	dt	dt	PROPN
ejde-761	144	1	=	=	SYM
ejde-761	144	2	∫	∫	PROPN
ejde-761	144	3	τ	τ	PROPN
ejde-761	144	4	0	0	NUM
ejde-761	144	5	∫	∫	PROPN
ejde-761	144	6	rn	rn	PROPN
ejde-761	144	7	|∆u|m∆u∆η	|∆u|m∆u∆η	NOUN
ejde-761	144	8	dx	dx	PROPN
ejde-761	144	9	dt	dt	PROPN
ejde-761	144	10	.	.	PUNCT
ejde-761	145	1	(	(	PUNCT
ejde-761	145	2	3.11	3.11	NUM
ejde-761	145	3	)	)	PUNCT
ejde-761	145	4	note	note	NOUN
ejde-761	145	5	that	that	SCONJ
ejde-761	145	6	under	under	ADP
ejde-761	145	7	the	the	DET
ejde-761	145	8	conditions	condition	NOUN
ejde-761	145	9	stated	state	VERB
ejde-761	145	10	in	in	ADP
ejde-761	145	11	(	(	PUNCT
ejde-761	145	12	3.10	3.10	NUM
ejde-761	145	13	)	)	PUNCT
ejde-761	145	14	,	,	PUNCT
ejde-761	145	15	proposition	proposition	NOUN
ejde-761	145	16	2.2	2.2	NUM
ejde-761	145	17	can	can	AUX
ejde-761	145	18	be	be	AUX
ejde-761	145	19	used	use	VERB
ejde-761	145	20	to	to	PART
ejde-761	145	21	further	far	ADV
ejde-761	145	22	develop	develop	VERB
ejde-761	145	23	the	the	DET
ejde-761	145	24	left	left	ADJ
ejde-761	145	25	-	-	PUNCT
ejde-761	145	26	hand	hand	NOUN
ejde-761	145	27	side	side	NOUN
ejde-761	145	28	integral	integral	ADJ
ejde-761	145	29	.	.	PUNCT
ejde-761	146	1	to	to	ADP
ejde-761	146	2	this	this	DET
ejde-761	146	3	end	end	NOUN
ejde-761	146	4	,	,	PUNCT
ejde-761	146	5	admit	admit	VERB
ejde-761	146	6	q	q	X
ejde-761	146	7	=	=	SYM
ejde-761	146	8	1	1	NUM
ejde-761	146	9	,	,	PUNCT
ejde-761	146	10	s	s	PART
ejde-761	146	11	=	=	SYM
ejde-761	146	12	2	2	NUM
ejde-761	146	13	and	and	CCONJ
ejde-761	146	14	α	α	NOUN
ejde-761	146	15	=	=	SYM
ejde-761	146	16	2	2	NUM
ejde-761	146	17	in	in	ADP
ejde-761	146	18	(	(	PUNCT
ejde-761	146	19	2.2	2.2	NUM
ejde-761	146	20	)	)	PUNCT
ejde-761	146	21	,	,	PUNCT
ejde-761	146	22	then∫	then∫	PROPN
ejde-761	146	23	rn	rn	PROPN
ejde-761	146	24	∆η∆u|∆u|m	∆η∆u|∆u|m	PROPN
ejde-761	146	25	dx	dx	PROPN
ejde-761	146	26	≤	≤	PROPN
ejde-761	147	1	k∥∆η∥1/2l2	k∥∆η∥1/2l2	PROPN
ejde-761	147	2	∥∇∆η∥1/2l2	∥∇∆η∥1/2l2	NUM
ejde-761	147	3	∥∇∆u∥1/2l2	∥∇∆u∥1/2l2	NUM
ejde-761	147	4	∥∆u∥m+1	∥∆u∥m+1	PROPN
ejde-761	147	5	l2	l2	NOUN
ejde-761	147	6	.	.	PUNCT
ejde-761	148	1	(	(	PUNCT
ejde-761	148	2	3.12	3.12	NUM
ejde-761	148	3	)	)	PUNCT
ejde-761	148	4	furthermore	furthermore	ADV
ejde-761	148	5	,	,	PUNCT
ejde-761	148	6	based	base	VERB
ejde-761	148	7	on	on	ADP
ejde-761	148	8	definition	definition	NOUN
ejde-761	148	9	2.6	2.6	NUM
ejde-761	148	10	and	and	CCONJ
ejde-761	148	11	lemma	lemma	PROPN
ejde-761	148	12	3.1	3.1	NUM
ejde-761	148	13	,	,	PUNCT
ejde-761	148	14	the	the	DET
ejde-761	148	15	following	follow	VERB
ejde-761	148	16	bounds	bound	NOUN
ejde-761	148	17	apply	apply	VERB
ejde-761	148	18	:	:	PUNCT
ejde-761	148	19	k∥∆η∥1/2l2	k∥∆η∥1/2l2	PROPN
ejde-761	148	20	∥∇∆η∥1/2l2	∥∇∆η∥1/2l2	NUM
ejde-761	148	21	∥∇∆u∥1/2l2	∥∇∆u∥1/2l2	NUM
ejde-761	148	22	∥∆u∥m+1	∥∆u∥m+1	PRON
ejde-761	148	23	l2	l2	NOUN
ejde-761	148	24	≤	≤	NUM
ejde-761	148	25	k∥η∥1/2	k∥η∥1/2	NOUN
ejde-761	148	26	h2	h2	PROPN
ejde-761	148	27	0	0	NUM
ejde-761	148	28	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	148	29	h3	h3	NOUN
ejde-761	148	30	0	0	NUM
ejde-761	148	31	∥u∥1/2	∥u∥1/2	PROPN
ejde-761	148	32	h3	h3	NOUN
ejde-761	148	33	0	0	NUM
ejde-761	148	34	∥u∥m+1	∥u∥m+1	X
ejde-761	148	35	h2	h2	NOUN
ejde-761	148	36	0	0	NUM
ejde-761	148	37	≤	≤	PROPN
ejde-761	148	38	k∥η∥1/2	k∥η∥1/2	NOUN
ejde-761	148	39	h2	h2	PROPN
ejde-761	148	40	0	0	NUM
ejde-761	148	41	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	148	42	h3	h3	NOUN
ejde-761	148	43	0	0	NUM
ejde-761	149	1	∥u0∥1/2h3	∥u0∥1/2h3	NUM
ejde-761	149	2	0	0	NUM
ejde-761	150	1	∥u0∥m+1	∥u0∥m+1	PROPN
ejde-761	150	2	h2	h2	PROPN
ejde-761	150	3	0	0	NUM
ejde-761	150	4	(	(	PUNCT
ejde-761	150	5	3.13	3.13	NUM
ejde-761	150	6	)	)	PUNCT
ejde-761	150	7	in	in	ADP
ejde-761	150	8	addition	addition	NOUN
ejde-761	150	9	,	,	PUNCT
ejde-761	150	10	∫	∫	PROPN
ejde-761	150	11	rn	rn	PROPN
ejde-761	150	12	|u|p−1uη	|u|p−1uη	PROPN
ejde-761	150	13	dx	dx	PROPN
ejde-761	150	14	≤	≤	NUM
ejde-761	150	15	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	150	16	∥∇η∥1/2l2	∥∇η∥1/2l2	PROPN
ejde-761	150	17	∥u∥1/2l2	∥u∥1/2l2	PROPN
ejde-761	150	18	∥∇u∥1/2l2	∥∇u∥1/2l2	PROPN
ejde-761	150	19	∥u∥p−1	∥u∥p−1	NOUN
ejde-761	150	20	l2	l2	NOUN
ejde-761	150	21	≤	≤	NOUN
ejde-761	150	22	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	150	23	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	150	24	h1	h1	PROPN
ejde-761	150	25	0	0	NUM
ejde-761	150	26	∥u∥1/2	∥u∥1/2	PROPN
ejde-761	150	27	h1	h1	PROPN
ejde-761	150	28	0	0	NUM
ejde-761	150	29	∥u∥p−	∥u∥p−	NOUN
ejde-761	150	30	1	1	NUM
ejde-761	150	31	2	2	NUM
ejde-761	150	32	l2	l2	NOUN
ejde-761	150	33	≤	≤	NOUN
ejde-761	150	34	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	150	35	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	150	36	h1	h1	PROPN
ejde-761	150	37	0	0	PROPN
ejde-761	150	38	∥u0∥1/2h1	∥u0∥1/2h1	PROPN
ejde-761	150	39	0	0	NUM
ejde-761	151	1	∥u0∥	∥u0∥	PROPN
ejde-761	151	2	p−	p−	VERB
ejde-761	151	3	1	1	NUM
ejde-761	151	4	2	2	NUM
ejde-761	151	5	l2	l2	NOUN
ejde-761	151	6	(	(	PUNCT
ejde-761	151	7	3.14	3.14	NUM
ejde-761	151	8	)	)	PUNCT
ejde-761	151	9	now	now	ADV
ejde-761	151	10	,	,	PUNCT
ejde-761	151	11	we	we	PRON
ejde-761	151	12	considering	consider	VERB
ejde-761	151	13	the	the	DET
ejde-761	151	14	natural	natural	ADJ
ejde-761	151	15	sobolev	sobolev	NOUN
ejde-761	151	16	embedding	embed	VERB
ejde-761	151	17	,	,	PUNCT
ejde-761	151	18	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	151	19	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	151	20	h1	h1	PROPN
ejde-761	151	21	0	0	PROPN
ejde-761	151	22	∥u0∥1/2h1	∥u0∥1/2h1	PROPN
ejde-761	151	23	0	0	NUM
ejde-761	151	24	∥u0∥	∥u0∥	PROPN
ejde-761	151	25	p−	p−	VERB
ejde-761	151	26	1	1	NUM
ejde-761	151	27	2	2	NUM
ejde-761	151	28	l2	l2	NOUN
ejde-761	151	29	≤	≤	NOUN
ejde-761	151	30	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	151	31	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	151	32	h1	h1	PROPN
ejde-761	151	33	0	0	NUM
ejde-761	151	34	∥u0∥ph1	∥u0∥ph1	PROPN
ejde-761	151	35	0	0	NUM
ejde-761	151	36	.	.	PUNCT
ejde-761	152	1	(	(	PUNCT
ejde-761	152	2	3.15	3.15	NUM
ejde-761	152	3	)	)	PUNCT
ejde-761	152	4	8	8	NUM
ejde-761	152	5	j.	j.	PROPN
ejde-761	152	6	l.	l.	PROPN
ejde-761	152	7	díaz	díaz	PROPN
ejde-761	152	8	palencia	palencia	PROPN
ejde-761	152	9	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	152	10	next	next	ADV
ejde-761	152	11	we	we	PRON
ejde-761	152	12	return	return	VERB
ejde-761	152	13	to	to	ADP
ejde-761	152	14	the	the	DET
ejde-761	152	15	first	first	ADJ
ejde-761	152	16	integral	integral	NOUN
ejde-761	152	17	in	in	ADP
ejde-761	152	18	(	(	PUNCT
ejde-761	152	19	3.11	3.11	NUM
ejde-761	152	20	)	)	PUNCT
ejde-761	152	21	,	,	PUNCT
ejde-761	152	22	and	and	CCONJ
ejde-761	152	23	by	by	ADP
ejde-761	152	24	using	use	VERB
ejde-761	152	25	the	the	DET
ejde-761	152	26	gronwall	gronwall	ADJ
ejde-761	152	27	inequality	inequality	NOUN
ejde-761	152	28	,	,	PUNCT
ejde-761	152	29	we	we	PRON
ejde-761	152	30	hav	hav	VERB
ejde-761	152	31	∫	∫	PROPN
ejde-761	152	32	τ	τ	PROPN
ejde-761	152	33	0	0	NUM
ejde-761	152	34	utη	utη	NOUN
ejde-761	152	35	dt	dt	PROPN
ejde-761	152	36	≤	≤	NUM
ejde-761	152	37	∫	∫	PROPN
ejde-761	152	38	τ	τ	PROPN
ejde-761	152	39	0	0	NUM
ejde-761	152	40	g(t)ηu	g(t)ηu	PROPN
ejde-761	152	41	dt	dt	PROPN
ejde-761	152	42	,	,	PUNCT
ejde-761	152	43	(	(	PUNCT
ejde-761	152	44	3.16	3.16	NUM
ejde-761	152	45	)	)	PUNCT
ejde-761	152	46	where	where	SCONJ
ejde-761	152	47	g(t	g(t	PROPN
ejde-761	152	48	)	)	PUNCT
ejde-761	152	49	is	be	AUX
ejde-761	152	50	a	a	DET
ejde-761	152	51	continuous	continuous	ADJ
ejde-761	152	52	function	function	NOUN
ejde-761	152	53	coming	come	VERB
ejde-761	152	54	from	from	ADP
ejde-761	152	55	the	the	DET
ejde-761	152	56	application	application	NOUN
ejde-761	152	57	of	of	ADP
ejde-761	152	58	the	the	DET
ejde-761	152	59	conditions	condition	NOUN
ejde-761	152	60	required	require	VERB
ejde-761	152	61	by	by	ADP
ejde-761	152	62	the	the	DET
ejde-761	152	63	gronwall	gronwall	ADJ
ejde-761	152	64	inequality	inequality	NOUN
ejde-761	152	65	.	.	PUNCT
ejde-761	153	1	note	note	VERB
ejde-761	153	2	that	that	SCONJ
ejde-761	153	3	the	the	DET
ejde-761	153	4	auxiliary	auxiliary	ADJ
ejde-761	153	5	function	function	PROPN
ejde-761	153	6	η	η	PROPN
ejde-761	153	7	is	be	AUX
ejde-761	153	8	c2(0	c2(0	PROPN
ejde-761	153	9	,	,	PUNCT
ejde-761	153	10	τ	τ	NOUN
ejde-761	153	11	)	)	PUNCT
ejde-761	153	12	as	as	SCONJ
ejde-761	153	13	expressed	express	VERB
ejde-761	153	14	in	in	ADP
ejde-761	153	15	(	(	PUNCT
ejde-761	153	16	3.10	3.10	NUM
ejde-761	153	17	)	)	PUNCT
ejde-761	153	18	.	.	PUNCT
ejde-761	154	1	then	then	ADV
ejde-761	154	2	,	,	PUNCT
ejde-761	154	3	η	η	PROPN
ejde-761	154	4	can	can	AUX
ejde-761	154	5	be	be	AUX
ejde-761	154	6	selected	select	VERB
ejde-761	154	7	such	such	ADJ
ejde-761	154	8	that	that	SCONJ
ejde-761	154	9	ηg(t	ηg(t	NOUN
ejde-761	154	10	)	)	PUNCT
ejde-761	154	11	≤	≤	NUM
ejde-761	154	12	1	1	NUM
ejde-761	154	13	.	.	PUNCT
ejde-761	154	14	consequently,∫	consequently,∫	NOUN
ejde-761	154	15	τ	τ	PROPN
ejde-761	154	16	0	0	NUM
ejde-761	154	17	utη	utη	NOUN
ejde-761	154	18	dt	dt	PROPN
ejde-761	154	19	≤	≤	NUM
ejde-761	154	20	∫	∫	PROPN
ejde-761	154	21	τ	τ	X
ejde-761	154	22	0	0	NUM
ejde-761	154	23	u	u	PROPN
ejde-761	154	24	dt	dt	PROPN
ejde-761	154	25	.	.	PUNCT
ejde-761	155	1	(	(	PUNCT
ejde-761	155	2	3.17	3.17	NUM
ejde-761	155	3	)	)	PUNCT
ejde-761	155	4	assuming	assume	VERB
ejde-761	155	5	that	that	SCONJ
ejde-761	155	6	the	the	DET
ejde-761	155	7	constant	constant	ADJ
ejde-761	155	8	k	k	X
ejde-761	155	9	in	in	ADP
ejde-761	155	10	(	(	PUNCT
ejde-761	155	11	3.13	3.13	NUM
ejde-761	155	12	)	)	PUNCT
ejde-761	155	13	is	be	AUX
ejde-761	155	14	sufficiently	sufficiently	ADV
ejde-761	155	15	large	large	ADJ
ejde-761	155	16	while	while	SCONJ
ejde-761	155	17	the	the	DET
ejde-761	155	18	constant	constant	ADJ
ejde-761	155	19	k	k	PROPN
ejde-761	155	20	is	be	AUX
ejde-761	155	21	considered	consider	VERB
ejde-761	155	22	sufficiently	sufficiently	ADV
ejde-761	155	23	small	small	ADJ
ejde-761	155	24	,	,	PUNCT
ejde-761	155	25	by	by	ADP
ejde-761	155	26	(	(	PUNCT
ejde-761	155	27	3.11),∫	3.11),∫	NUM
ejde-761	155	28	τ	τ	X
ejde-761	155	29	0	0	NUM
ejde-761	155	30	∥u∥h4dt+	∥u∥h4dt+	ADV
ejde-761	156	1	k	k	PRON
ejde-761	156	2	∫	∫	PROPN
ejde-761	156	3	τ	τ	PROPN
ejde-761	156	4	0	0	NUM
ejde-761	156	5	∥η∥1/2l2	∥η∥1/2l2	PROPN
ejde-761	156	6	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	156	7	h1	h1	PROPN
ejde-761	156	8	0	0	NUM
ejde-761	156	9	∥u0∥ph1	∥u0∥ph1	PROPN
ejde-761	156	10	0	0	NUM
ejde-761	156	11	dt	dt	NOUN
ejde-761	156	12	≤	≤	PROPN
ejde-761	157	1	k	k	PROPN
ejde-761	157	2	∫	∫	PROPN
ejde-761	157	3	τ	τ	PROPN
ejde-761	157	4	0	0	NUM
ejde-761	157	5	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	157	6	h2	h2	PROPN
ejde-761	157	7	0	0	NUM
ejde-761	157	8	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	157	9	h3	h3	NOUN
ejde-761	157	10	0	0	NUM
ejde-761	157	11	∥u0∥1/2h3	∥u0∥1/2h3	NUM
ejde-761	157	12	0	0	NUM
ejde-761	158	1	∥u0∥m+1	∥u0∥m+1	PROPN
ejde-761	158	2	h2	h2	PROPN
ejde-761	158	3	0	0	NUM
ejde-761	158	4	dt	dt	PROPN
ejde-761	158	5	.	.	PUNCT
ejde-761	159	1	(	(	PUNCT
ejde-761	159	2	3.18	3.18	NUM
ejde-761	159	3	)	)	PUNCT
ejde-761	159	4	note	note	NOUN
ejde-761	159	5	that	that	SCONJ
ejde-761	159	6	the	the	DET
ejde-761	159	7	functions	function	NOUN
ejde-761	159	8	η	η	PROPN
ejde-761	159	9	and	and	CCONJ
ejde-761	159	10	u0	u0	PROPN
ejde-761	159	11	satisfy	satisfy	VERB
ejde-761	159	12	the	the	DET
ejde-761	159	13	conditions	condition	NOUN
ejde-761	159	14	expressed	express	VERB
ejde-761	159	15	in	in	ADP
ejde-761	159	16	(	(	PUNCT
ejde-761	159	17	3.10	3.10	NUM
ejde-761	159	18	)	)	PUNCT
ejde-761	159	19	,	,	PUNCT
ejde-761	159	20	in	in	ADP
ejde-761	159	21	particular	particular	ADJ
ejde-761	159	22	the	the	DET
ejde-761	159	23	compact	compact	ADJ
ejde-761	159	24	support	support	NOUN
ejde-761	159	25	.	.	PUNCT
ejde-761	160	1	consequently	consequently	ADV
ejde-761	160	2	and	and	CCONJ
ejde-761	160	3	locally	locally	ADV
ejde-761	160	4	in	in	ADP
ejde-761	160	5	time	time	NOUN
ejde-761	160	6	(	(	PUNCT
ejde-761	160	7	i.e.	i.e.	X
ejde-761	160	8	locally	locally	ADV
ejde-761	160	9	in	in	ADP
ejde-761	160	10	the	the	DET
ejde-761	160	11	proximity	proximity	NOUN
ejde-761	160	12	of	of	ADP
ejde-761	160	13	a	a	DET
ejde-761	160	14	sufficiently	sufficiently	ADV
ejde-761	160	15	small	small	ADJ
ejde-761	160	16	τ	τ	PUNCT
ejde-761	160	17	to	to	PART
ejde-761	160	18	keep	keep	VERB
ejde-761	160	19	the	the	DET
ejde-761	160	20	support	support	NOUN
ejde-761	160	21	):	):	PUNCT
ejde-761	160	22	∥u∥h4	∥u∥h4	NOUN
ejde-761	160	23	0	0	X
ejde-761	161	1	+	+	NUM
ejde-761	161	2	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	161	3	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	161	4	h1	h1	PROPN
ejde-761	161	5	0	0	NUM
ejde-761	161	6	∥u0∥ph1	∥u0∥ph1	PROPN
ejde-761	161	7	0	0	NUM
ejde-761	161	8	≤	≤	NOUN
ejde-761	161	9	k∥η∥1/2	k∥η∥1/2	NOUN
ejde-761	161	10	h2	h2	PROPN
ejde-761	161	11	0	0	NUM
ejde-761	161	12	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	161	13	h3	h3	NOUN
ejde-761	161	14	0	0	NUM
ejde-761	161	15	∥u0∥1/2h3	∥u0∥1/2h3	NUM
ejde-761	161	16	0	0	NUM
ejde-761	162	1	∥u0∥m+1	∥u0∥m+1	PROPN
ejde-761	162	2	h2	h2	PROPN
ejde-761	162	3	0	0	NUM
ejde-761	162	4	,	,	PUNCT
ejde-761	162	5	(	(	PUNCT
ejde-761	162	6	3.19	3.19	NUM
ejde-761	162	7	)	)	PUNCT
ejde-761	162	8	which	which	PRON
ejde-761	162	9	permits	permit	VERB
ejde-761	162	10	to	to	PART
ejde-761	162	11	account	account	VERB
ejde-761	162	12	for	for	ADP
ejde-761	162	13	the	the	DET
ejde-761	162	14	bound	bind	VERB
ejde-761	162	15	properties	property	NOUN
ejde-761	162	16	of	of	ADP
ejde-761	162	17	any	any	DET
ejde-761	162	18	solution	solution	NOUN
ejde-761	162	19	in	in	ADP
ejde-761	162	20	h4	h4	PROPN
ejde-761	162	21	.	.	PUNCT
ejde-761	163	1	to	to	ADP
ejde-761	163	2	this	this	DET
ejde-761	163	3	end	end	NOUN
ejde-761	163	4	,	,	PUNCT
ejde-761	163	5	it	it	PRON
ejde-761	163	6	suffices	suffice	VERB
ejde-761	163	7	to	to	PART
ejde-761	163	8	write	write	VERB
ejde-761	163	9	∥u∥h4	∥u∥h4	NOUN
ejde-761	163	10	0	0	NUM
ejde-761	163	11	≤	≤	PROPN
ejde-761	163	12	k∥η∥1/2	k∥η∥1/2	NOUN
ejde-761	163	13	h2	h2	PROPN
ejde-761	163	14	0	0	NUM
ejde-761	163	15	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	163	16	h3	h3	NOUN
ejde-761	163	17	0	0	NUM
ejde-761	164	1	∥u0∥1/2h3	∥u0∥1/2h3	NUM
ejde-761	164	2	0	0	NUM
ejde-761	165	1	∥u0∥m+1	∥u0∥m+1	PROPN
ejde-761	165	2	h2	h2	PROPN
ejde-761	165	3	0	0	NUM
ejde-761	166	1	+	+	CCONJ
ejde-761	166	2	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	166	3	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	166	4	h1	h1	PROPN
ejde-761	166	5	0	0	NUM
ejde-761	166	6	∥u0∥ph1	∥u0∥ph1	PROPN
ejde-761	166	7	0	0	NUM
ejde-761	166	8	,	,	PUNCT
ejde-761	166	9	(	(	PUNCT
ejde-761	166	10	3.20	3.20	NUM
ejde-761	166	11	)	)	PUNCT
ejde-761	166	12	which	which	PRON
ejde-761	166	13	allows	allow	VERB
ejde-761	166	14	us	we	PRON
ejde-761	166	15	showing	show	VERB
ejde-761	166	16	the	the	DET
ejde-761	166	17	lemma	lemma	PROPN
ejde-761	166	18	postulations	postulations	PROPN
ejde-761	166	19	.	.	PUNCT
ejde-761	167	1	□	□	PUNCT
ejde-761	167	2	the	the	DET
ejde-761	167	3	next	next	ADJ
ejde-761	167	4	lemma	lemma	PROPN
ejde-761	167	5	aims	aim	VERB
ejde-761	167	6	at	at	ADP
ejde-761	167	7	exploring	explore	VERB
ejde-761	167	8	the	the	DET
ejde-761	167	9	bound	bind	VERB
ejde-761	167	10	of	of	ADP
ejde-761	167	11	oscillating	oscillate	VERB
ejde-761	167	12	solutions	solution	NOUN
ejde-761	167	13	by	by	ADP
ejde-761	167	14	the	the	DET
ejde-761	167	15	defined	define	VERB
ejde-761	167	16	mollifier	mollifier	NOUN
ejde-761	167	17	in	in	ADP
ejde-761	167	18	(	(	PUNCT
ejde-761	167	19	2.6	2.6	NUM
ejde-761	167	20	)	)	PUNCT
ejde-761	167	21	.	.	PUNCT
ejde-761	168	1	to	to	ADP
ejde-761	168	2	this	this	DET
ejde-761	168	3	end	end	NOUN
ejde-761	168	4	,	,	PUNCT
ejde-761	168	5	the	the	DET
ejde-761	168	6	support	support	NOUN
ejde-761	168	7	is	be	AUX
ejde-761	168	8	kept	keep	VERB
ejde-761	168	9	free	free	ADJ
ejde-761	168	10	and	and	CCONJ
ejde-761	168	11	oscillating	oscillating	NOUN
ejde-761	168	12	.	.	PUNCT
ejde-761	169	1	as	as	ADP
ejde-761	169	2	a	a	DET
ejde-761	169	3	consequence	consequence	NOUN
ejde-761	169	4	,	,	PUNCT
ejde-761	169	5	the	the	DET
ejde-761	169	6	following	follow	VERB
ejde-761	169	7	is	be	AUX
ejde-761	169	8	required	require	VERB
ejde-761	169	9	previously	previously	ADV
ejde-761	169	10	,	,	PUNCT
ejde-761	169	11	η	η	PROPN
ejde-761	169	12	∈	∈	PROPN
ejde-761	169	13	l1(0	l1(0	PROPN
ejde-761	169	14	,	,	PUNCT
ejde-761	169	15	τ	τ	PROPN
ejde-761	169	16	;	;	PUNCT
ejde-761	169	17	h	h	PROPN
ejde-761	169	18	m	m	PROPN
ejde-761	169	19	ρ	ρ	PROPN
ejde-761	169	20	(	(	PUNCT
ejde-761	169	21	rn	rn	PROPN
ejde-761	169	22	)	)	PUNCT
ejde-761	169	23	)	)	PUNCT
ejde-761	169	24	∩	∩	PROPN
ejde-761	169	25	c2(0	c2(0	PROPN
ejde-761	169	26	,	,	PUNCT
ejde-761	169	27	τ	τ	X
ejde-761	169	28	;	;	PUNCT
ejde-761	169	29	hm	hm	INTJ
ejde-761	169	30	ρ	ρ	PROPN
ejde-761	169	31	∩	∩	PROPN
ejde-761	169	32	c	c	PROPN
ejde-761	169	33	n(rn	n(rn	PROPN
ejde-761	169	34	)	)	PUNCT
ejde-761	169	35	)	)	PUNCT
ejde-761	169	36	,	,	PUNCT
ejde-761	169	37	u0(x	u0(x	X
ejde-761	169	38	)	)	PUNCT
ejde-761	169	39	∈	∈	NOUN
ejde-761	169	40	hm	hm	INTJ
ejde-761	169	41	ρ	ρ	PROPN
ejde-761	169	42	(	(	PUNCT
ejde-761	169	43	rn	rn	PROPN
ejde-761	169	44	)	)	PUNCT
ejde-761	170	1	∩	∩	PROPN
ejde-761	170	2	c	c	PROPN
ejde-761	170	3	n(rn	n(rn	PROPN
ejde-761	170	4	)	)	PUNCT
ejde-761	170	5	,	,	PUNCT
ejde-761	170	6	(	(	PUNCT
ejde-761	170	7	3.21	3.21	NUM
ejde-761	170	8	)	)	PUNCT
ejde-761	170	9	lemma	lemma	PROPN
ejde-761	170	10	3.3	3.3	NUM
ejde-761	170	11	.	.	PUNCT
ejde-761	171	1	each	each	PRON
ejde-761	171	2	oscillating	oscillate	VERB
ejde-761	171	3	energy	energy	NOUN
ejde-761	171	4	solution	solution	NOUN
ejde-761	171	5	satisfying	satisfying	ADJ
ejde-761	171	6	(	(	PUNCT
ejde-761	171	7	2.1	2.1	NUM
ejde-761	171	8	)	)	PUNCT
ejde-761	171	9	is	be	AUX
ejde-761	171	10	globally	globally	ADV
ejde-761	171	11	bounded	bound	VERB
ejde-761	171	12	by	by	ADP
ejde-761	171	13	the	the	DET
ejde-761	171	14	mollification	mollification	NOUN
ejde-761	171	15	introduced	introduce	VERB
ejde-761	171	16	in	in	ADP
ejde-761	171	17	the	the	DET
ejde-761	171	18	norm	norm	NOUN
ejde-761	171	19	(	(	PUNCT
ejde-761	171	20	2.6	2.6	NUM
ejde-761	171	21	)	)	PUNCT
ejde-761	171	22	.	.	PUNCT
ejde-761	172	1	proof	proof	NOUN
ejde-761	172	2	.	.	PUNCT
ejde-761	173	1	to	to	PART
ejde-761	173	2	show	show	VERB
ejde-761	173	3	the	the	DET
ejde-761	173	4	proposed	propose	VERB
ejde-761	173	5	lemma	lemma	PROPN
ejde-761	173	6	,	,	PUNCT
ejde-761	173	7	we	we	PRON
ejde-761	173	8	use	use	VERB
ejde-761	173	9	(	(	PUNCT
ejde-761	173	10	3.11	3.11	NUM
ejde-761	173	11	)	)	PUNCT
ejde-761	173	12	and	and	CCONJ
ejde-761	173	13	operate	operate	VERB
ejde-761	173	14	with	with	ADP
ejde-761	173	15	the	the	DET
ejde-761	173	16	inequality	inequality	NOUN
ejde-761	173	17	(	(	PUNCT
ejde-761	173	18	2.2	2.2	NUM
ejde-761	173	19	)	)	PUNCT
ejde-761	173	20	(	(	PUNCT
ejde-761	173	21	with	with	ADP
ejde-761	173	22	q	q	NOUN
ejde-761	173	23	=	=	SYM
ejde-761	173	24	1	1	NUM
ejde-761	173	25	,	,	PUNCT
ejde-761	173	26	s	s	PART
ejde-761	173	27	=	=	SYM
ejde-761	173	28	2	2	NUM
ejde-761	173	29	,	,	PUNCT
ejde-761	173	30	α	α	NOUN
ejde-761	173	31	=	=	SYM
ejde-761	173	32	2	2	NUM
ejde-761	173	33	)	)	PUNCT
ejde-761	173	34	along	along	ADP
ejde-761	173	35	with	with	ADP
ejde-761	173	36	the	the	DET
ejde-761	173	37	norm	norm	NOUN
ejde-761	173	38	(	(	PUNCT
ejde-761	173	39	2.3	2.3	NUM
ejde-761	173	40	)	)	PUNCT
ejde-761	173	41	.	.	PUNCT
ejde-761	174	1	then∫	then∫	PROPN
ejde-761	174	2	rn	rn	PROPN
ejde-761	174	3	∆η∆u	∆η∆u	PROPN
ejde-761	174	4	|∆u|m	|∆u|m	NOUN
ejde-761	174	5	dx	dx	PROPN
ejde-761	174	6	≤	≤	PROPN
ejde-761	174	7	k∥∆η∥1/2l2	k∥∆η∥1/2l2	PROPN
ejde-761	174	8	∥∇∆η∥1/2l2	∥∇∆η∥1/2l2	NUM
ejde-761	174	9	∥∇∆u∥1/2l2	∥∇∆u∥1/2l2	NUM
ejde-761	174	10	∥∆u∥m+1	∥∆u∥m+1	PROPN
ejde-761	174	11	l2	l2	NOUN
ejde-761	174	12	≤	≤	NUM
ejde-761	174	13	k∥η∥1/2	k∥η∥1/2	NOUN
ejde-761	174	14	θ	θ	PROPN
ejde-761	174	15	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	174	16	θ	θ	PROPN
ejde-761	175	1	∥u∥1/2	∥u∥1/2	NOUN
ejde-761	175	2	θ	θ	PROPN
ejde-761	175	3	∥u∥m+1	∥u∥m+1	X
ejde-761	175	4	θ	θ	X
ejde-761	175	5	.	.	PUNCT
ejde-761	176	1	(	(	PUNCT
ejde-761	176	2	3.22	3.22	NUM
ejde-761	176	3	)	)	PUNCT
ejde-761	176	4	now	now	ADV
ejde-761	176	5	,	,	PUNCT
ejde-761	176	6	based	base	VERB
ejde-761	176	7	on	on	ADP
ejde-761	176	8	lemma	lemma	PROPN
ejde-761	176	9	3.1	3.1	NUM
ejde-761	176	10	,	,	PUNCT
ejde-761	176	11	the	the	DET
ejde-761	176	12	following	follow	VERB
ejde-761	176	13	bound	bind	VERB
ejde-761	176	14	in	in	ADP
ejde-761	176	15	hm	hm	INTJ
ejde-761	176	16	ρ	ρ	PROPN
ejde-761	176	17	is	be	AUX
ejde-761	176	18	obtained	obtain	VERB
ejde-761	176	19	.	.	PUNCT
ejde-761	177	1	k∥η∥1/2	k∥η∥1/2	NOUN
ejde-761	177	2	θ	θ	PROPN
ejde-761	177	3	∥η∥1/2	∥η∥1/2	PROPN
ejde-761	177	4	θ	θ	PROPN
ejde-761	177	5	∥u∥1/2	∥u∥1/2	NOUN
ejde-761	177	6	θ	θ	PROPN
ejde-761	177	7	∥u∥m+1	∥u∥m+1	NOUN
ejde-761	177	8	θ	θ	PROPN
ejde-761	177	9	≤	≤	PROPN
ejde-761	177	10	kσ∥η∥hm	kσ∥η∥hm	PROPN
ejde-761	177	11	ρ	ρ	PROPN
ejde-761	177	12	∥u0∥	∥u0∥	PROPN
ejde-761	177	13	m+	m+	NUM
ejde-761	177	14	3	3	NUM
ejde-761	177	15	2	2	NUM
ejde-761	177	16	hm	hm	INTJ
ejde-761	177	17	ρ	ρ	NOUN
ejde-761	177	18	.	.	PUNCT
ejde-761	178	1	(	(	PUNCT
ejde-761	178	2	3.23	3.23	NUM
ejde-761	178	3	)	)	PUNCT
ejde-761	178	4	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	178	5	instability	instability	NOUN
ejde-761	178	6	of	of	ADP
ejde-761	178	7	energy	energy	NOUN
ejde-761	178	8	solutions	solution	NOUN
ejde-761	178	9	9	9	NUM
ejde-761	178	10	operating	operate	VERB
ejde-761	178	11	similarly,∫	similarly,∫	ADJ
ejde-761	178	12	rn	rn	PROPN
ejde-761	178	13	|u|p−1uη	|u|p−1uη	PROPN
ejde-761	178	14	dx	dx	PROPN
ejde-761	178	15	≤	≤	NUM
ejde-761	178	16	k∥η∥1/2l2	k∥η∥1/2l2	PROPN
ejde-761	178	17	∥∇η∥1/2l2	∥∇η∥1/2l2	PROPN
ejde-761	178	18	∥u∥1/2l2	∥u∥1/2l2	PROPN
ejde-761	178	19	∥∇u∥1/2l2	∥∇u∥1/2l2	PROPN
ejde-761	178	20	∥u∥p−1	∥u∥p−1	NOUN
ejde-761	178	21	l2	l2	VERB
ejde-761	178	22	≤	≤	NUM
ejde-761	178	23	k∥η∥1/2θ	k∥η∥1/2θ	PROPN
ejde-761	178	24	∥η∥1/2θ	∥η∥1/2θ	PRON
ejde-761	178	25	∥u∥1/2θ	∥u∥1/2θ	PROPN
ejde-761	178	26	∥u∥p−	∥u∥p−	NOUN
ejde-761	178	27	1	1	NUM
ejde-761	178	28	2	2	NUM
ejde-761	178	29	θ	θ	PROPN
ejde-761	178	30	≤	≤	NOUN
ejde-761	178	31	kσ∥η∥hm	kσ∥η∥hm	PROPN
ejde-761	178	32	ρ	ρ	PROPN
ejde-761	178	33	∥u0∥phm	∥u0∥phm	PROPN
ejde-761	178	34	ρ	ρ	PROPN
ejde-761	178	35	(	(	PUNCT
ejde-761	178	36	3.24	3.24	NUM
ejde-761	178	37	)	)	PUNCT
ejde-761	178	38	now	now	ADV
ejde-761	178	39	,	,	PUNCT
ejde-761	178	40	considering	consider	VERB
ejde-761	178	41	(	(	PUNCT
ejde-761	178	42	3.11	3.11	NUM
ejde-761	178	43	)	)	PUNCT
ejde-761	178	44	,	,	PUNCT
ejde-761	178	45	applying	apply	VERB
ejde-761	178	46	the	the	DET
ejde-761	178	47	gronwall	gronwall	ADJ
ejde-761	178	48	inequality	inequality	NOUN
ejde-761	178	49	similarly	similarly	ADV
ejde-761	178	50	as	as	ADP
ejde-761	178	51	in	in	ADP
ejde-761	178	52	(	(	PUNCT
ejde-761	178	53	3.17	3.17	NUM
ejde-761	178	54	)	)	PUNCT
ejde-761	178	55	,	,	PUNCT
ejde-761	178	56	the	the	DET
ejde-761	178	57	following	follow	VERB
ejde-761	178	58	inequality	inequality	NOUN
ejde-761	178	59	holds	hold	VERB
ejde-761	178	60	for	for	ADP
ejde-761	178	61	k	k	NOUN
ejde-761	178	62	sufficiently	sufficiently	ADV
ejde-761	178	63	large	large	ADJ
ejde-761	178	64	and	and	CCONJ
ejde-761	178	65	k	k	PROPN
ejde-761	178	66	small,∫	small,∫	NOUN
ejde-761	178	67	τ	τ	PROPN
ejde-761	178	68	0	0	NUM
ejde-761	178	69	∥u∥θdt+	∥u∥θdt+	PUNCT
ejde-761	179	1	kσ	kσ	PROPN
ejde-761	179	2	∫	∫	PROPN
ejde-761	179	3	τ	τ	PROPN
ejde-761	179	4	0	0	PROPN
ejde-761	179	5	∥η∥hm	∥η∥hm	NOUN
ejde-761	179	6	ρ	ρ	X
ejde-761	179	7	∥u0∥phm	∥u0∥phm	PROPN
ejde-761	179	8	ρ	ρ	PROPN
ejde-761	179	9	dt	dt	NOUN
ejde-761	179	10	≤	≤	NUM
ejde-761	179	11	kσ	kσ	PROPN
ejde-761	179	12	∫	∫	PROPN
ejde-761	179	13	τ	τ	PROPN
ejde-761	179	14	0	0	NUM
ejde-761	179	15	∥η∥hm	∥η∥hm	NOUN
ejde-761	179	16	ρ	ρ	PROPN
ejde-761	179	17	∥u0∥	∥u0∥	PROPN
ejde-761	179	18	m+	m+	NUM
ejde-761	179	19	3	3	NUM
ejde-761	179	20	2	2	NUM
ejde-761	179	21	hm	hm	INTJ
ejde-761	179	22	ρ	ρ	NUM
ejde-761	179	23	dt	dt	PROPN
ejde-761	179	24	.	.	PUNCT
ejde-761	180	1	(	(	PUNCT
ejde-761	180	2	3.25	3.25	NUM
ejde-761	180	3	)	)	PUNCT
ejde-761	180	4	note	note	VERB
ejde-761	180	5	that	that	SCONJ
ejde-761	180	6	the	the	DET
ejde-761	180	7	functions	function	NOUN
ejde-761	180	8	η	η	PROPN
ejde-761	180	9	and	and	CCONJ
ejde-761	180	10	u0	u0	PROPN
ejde-761	180	11	satisfy	satisfy	NOUN
ejde-761	180	12	(	(	PUNCT
ejde-761	180	13	3.21	3.21	NUM
ejde-761	180	14	)	)	PUNCT
ejde-761	180	15	.	.	PUNCT
ejde-761	181	1	then	then	ADV
ejde-761	181	2	any	any	DET
ejde-761	181	3	oscillating	oscillate	VERB
ejde-761	181	4	solution	solution	NOUN
ejde-761	181	5	,	,	PUNCT
ejde-761	181	6	under	under	ADP
ejde-761	181	7	the	the	DET
ejde-761	181	8	norm	norm	NOUN
ejde-761	181	9	(	(	PUNCT
ejde-761	181	10	2.3	2.3	NUM
ejde-761	181	11	)	)	PUNCT
ejde-761	181	12	,	,	PUNCT
ejde-761	181	13	in	in	ADP
ejde-761	181	14	(	(	PUNCT
ejde-761	181	15	0	0	NUM
ejde-761	181	16	,	,	PUNCT
ejde-761	181	17	τ	τ	X
ejde-761	181	18	)	)	PUNCT
ejde-761	181	19	is	be	AUX
ejde-761	181	20	mollified	mollify	VERB
ejde-761	181	21	by	by	ADP
ejde-761	181	22	the	the	DET
ejde-761	181	23	norm	norm	NOUN
ejde-761	181	24	(	(	PUNCT
ejde-761	181	25	2.6	2.6	NUM
ejde-761	181	26	)	)	PUNCT
ejde-761	181	27	.	.	PUNCT
ejde-761	182	1	based	base	VERB
ejde-761	182	2	on	on	ADP
ejde-761	182	3	this	this	PRON
ejde-761	182	4	,	,	PUNCT
ejde-761	182	5	the	the	DET
ejde-761	182	6	following	follow	VERB
ejde-761	182	7	bound	bind	VERB
ejde-761	182	8	holds	hold	NOUN
ejde-761	182	9	,	,	PUNCT
ejde-761	182	10	∥u∥θ	∥u∥θ	NOUN
ejde-761	182	11	≤	≤	ADJ
ejde-761	183	1	kσ∥η∥hm	kσ∥η∥hm	PROPN
ejde-761	183	2	ρ	ρ	PROPN
ejde-761	183	3	∥u0∥phm	∥u0∥phm	PROPN
ejde-761	183	4	ρ	ρ	PROPN
ejde-761	184	1	+	+	PROPN
ejde-761	184	2	kσ∥η∥hm	kσ∥η∥hm	PROPN
ejde-761	184	3	ρ	ρ	PROPN
ejde-761	184	4	∥u0∥	∥u0∥	PROPN
ejde-761	184	5	m+	m+	NUM
ejde-761	184	6	3	3	NUM
ejde-761	184	7	2	2	NUM
ejde-761	184	8	hm	hm	INTJ
ejde-761	184	9	ρ	ρ	NOUN
ejde-761	184	10	.	.	PUNCT
ejde-761	185	1	(	(	PUNCT
ejde-761	185	2	3.26	3.26	NUM
ejde-761	185	3	)	)	PUNCT
ejde-761	185	4	□	□	SYM
ejde-761	185	5	4	4	X
ejde-761	185	6	.	.	X
ejde-761	185	7	travelling	travel	VERB
ejde-761	185	8	waves	wave	NOUN
ejde-761	185	9	the	the	DET
ejde-761	185	10	travelling	travel	VERB
ejde-761	185	11	waves	wave	NOUN
ejde-761	185	12	(	(	PUNCT
ejde-761	185	13	tw	tw	NOUN
ejde-761	185	14	)	)	PUNCT
ejde-761	185	15	solutions	solution	NOUN
ejde-761	185	16	are	be	AUX
ejde-761	185	17	given	give	VERB
ejde-761	185	18	by	by	ADP
ejde-761	185	19	the	the	DET
ejde-761	185	20	change	change	NOUN
ejde-761	185	21	u(x	u(x	NOUN
ejde-761	185	22	,	,	PUNCT
ejde-761	185	23	t	t	PROPN
ejde-761	185	24	)	)	PUNCT
ejde-761	185	25	=	=	SYM
ejde-761	185	26	γ(ξ	γ(ξ	PROPN
ejde-761	185	27	)	)	PUNCT
ejde-761	185	28	where	where	SCONJ
ejde-761	185	29	ξ	ξ	X
ejde-761	185	30	=	=	SYM
ejde-761	185	31	x	x	X
ejde-761	185	32	·	·	PUNCT
ejde-761	185	33	nd	nd	INTJ
ejde-761	185	34	−	−	NOUN
ejde-761	185	35	λt	λt	ADP
ejde-761	185	36	.	.	PROPN
ejde-761	185	37	note	note	VERB
ejde-761	185	38	that	that	SCONJ
ejde-761	185	39	ξ	ξ	PROPN
ejde-761	185	40	∈	∈	PROPN
ejde-761	185	41	r	r	NOUN
ejde-761	185	42	and	and	CCONJ
ejde-761	185	43	the	the	DET
ejde-761	185	44	vector	vector	NOUN
ejde-761	185	45	nd	nd	PROPN
ejde-761	185	46	∈	∈	PROPN
ejde-761	185	47	rn	rn	PROPN
ejde-761	185	48	represents	represent	VERB
ejde-761	185	49	the	the	DET
ejde-761	185	50	tw	tw	NOUN
ejde-761	185	51	propagating	propagate	VERB
ejde-761	185	52	direction	direction	NOUN
ejde-761	185	53	.	.	PUNCT
ejde-761	186	1	in	in	ADP
ejde-761	186	2	addition	addition	NOUN
ejde-761	186	3	,	,	PUNCT
ejde-761	186	4	note	note	VERB
ejde-761	186	5	that	that	SCONJ
ejde-761	186	6	λ	λ	NOUN
ejde-761	186	7	represents	represent	VERB
ejde-761	186	8	the	the	DET
ejde-761	186	9	tw	tw	NOUN
ejde-761	186	10	velocity	velocity	NOUN
ejde-761	186	11	and	and	CCONJ
ejde-761	186	12	γ	γ	X
ejde-761	186	13	:	:	PUNCT
ejde-761	186	14	r	r	NOUN
ejde-761	186	15	→	→	SYM
ejde-761	186	16	(	(	PUNCT
ejde-761	186	17	0,∞	0,∞	NUM
ejde-761	186	18	)	)	PUNCT
ejde-761	186	19	is	be	AUX
ejde-761	186	20	the	the	DET
ejde-761	186	21	tw	tw	NOUN
ejde-761	186	22	profile	profile	NOUN
ejde-761	186	23	that	that	PRON
ejde-761	186	24	complies	comply	VERB
ejde-761	186	25	with	with	ADP
ejde-761	186	26	the	the	DET
ejde-761	186	27	norm	norm	NOUN
ejde-761	186	28	(	(	PUNCT
ejde-761	186	29	2.3	2.3	NUM
ejde-761	186	30	)	)	PUNCT
ejde-761	186	31	i.e.	i.e.	X
ejde-761	186	32	γ	γ	X
ejde-761	186	33	∈	∈	PROPN
ejde-761	186	34	hθ(r	hθ(r	NOUN
ejde-761	186	35	)	)	PUNCT
ejde-761	186	36	⊂	⊂	PROPN
ejde-761	186	37	l2	l2	PROPN
ejde-761	186	38	θ(r	θ(r	NOUN
ejde-761	186	39	)	)	PUNCT
ejde-761	186	40	⊂	⊂	PROPN
ejde-761	186	41	l2(r	l2(r	NOUN
ejde-761	186	42	)	)	PUNCT
ejde-761	186	43	.	.	PUNCT
ejde-761	187	1	it	it	PRON
ejde-761	187	2	is	be	AUX
ejde-761	187	3	to	to	PART
ejde-761	187	4	be	be	AUX
ejde-761	187	5	noted	note	VERB
ejde-761	187	6	that	that	SCONJ
ejde-761	187	7	two	two	NUM
ejde-761	187	8	tws	tws	NOUN
ejde-761	187	9	are	be	AUX
ejde-761	187	10	equivalent	equivalent	ADJ
ejde-761	187	11	under	under	ADP
ejde-761	187	12	discrete	discrete	ADJ
ejde-761	187	13	symmetry	symmetry	NOUN
ejde-761	187	14	(	(	PUNCT
ejde-761	187	15	ξ	ξ	PROPN
ejde-761	187	16	→	→	SYM
ejde-761	187	17	−ξ	−ξ	NOUN
ejde-761	187	18	)	)	PUNCT
ejde-761	187	19	and	and	CCONJ
ejde-761	187	20	translation	translation	NOUN
ejde-761	187	21	(	(	PUNCT
ejde-761	187	22	ξ	ξ	X
ejde-761	187	23	→	→	SYM
ejde-761	187	24	ξ	ξ	PROPN
ejde-761	187	25	+	+	SYM
ejde-761	187	26	ξ0	ξ0	NUM
ejde-761	187	27	)	)	PUNCT
ejde-761	187	28	.	.	PUNCT
ejde-761	188	1	we	we	PRON
ejde-761	188	2	assume	assume	VERB
ejde-761	188	3	that	that	SCONJ
ejde-761	188	4	the	the	DET
ejde-761	188	5	tw	tw	PROPN
ejde-761	188	6	direction	direction	NOUN
ejde-761	188	7	of	of	ADP
ejde-761	188	8	motion	motion	NOUN
ejde-761	188	9	is	be	AUX
ejde-761	188	10	nd	nd	ADP
ejde-761	188	11	=	=	PUNCT
ejde-761	188	12	(	(	PUNCT
ejde-761	188	13	1	1	NUM
ejde-761	188	14	,	,	PUNCT
ejde-761	188	15	0	0	NUM
ejde-761	188	16	,	,	PUNCT
ejde-761	188	17	0	0	NUM
ejde-761	188	18	,	,	PUNCT
ejde-761	188	19	.	.	PUNCT
ejde-761	188	20	.	.	PUNCT
ejde-761	188	21	.	.	PUNCT
ejde-761	189	1	,	,	PUNCT
ejde-761	189	2	0	0	NUM
ejde-761	189	3	)	)	PUNCT
ejde-761	189	4	,	,	PUNCT
ejde-761	189	5	such	such	ADJ
ejde-761	189	6	that	that	SCONJ
ejde-761	189	7	ξ	ξ	X
ejde-761	189	8	=	=	PUNCT
ejde-761	189	9	x	x	SYM
ejde-761	189	10	−	−	NOUN
ejde-761	189	11	λt	λt	X
ejde-761	189	12	and	and	CCONJ
ejde-761	189	13	u(x	u(x	NOUN
ejde-761	189	14	,	,	PUNCT
ejde-761	189	15	t	t	PROPN
ejde-761	189	16	)	)	PUNCT
ejde-761	189	17	=	=	SYM
ejde-761	189	18	γ(ξ	γ(ξ	PROPN
ejde-761	189	19	)	)	PUNCT
ejde-761	190	1	∈	∈	PROPN
ejde-761	190	2	r.	r.	NOUN
ejde-761	190	3	using	use	VERB
ejde-761	190	4	the	the	DET
ejde-761	190	5	described	describe	VERB
ejde-761	190	6	transformation	transformation	NOUN
ejde-761	190	7	into	into	ADP
ejde-761	190	8	the	the	DET
ejde-761	190	9	tw	tw	NOUN
ejde-761	190	10	-	-	NOUN
ejde-761	190	11	domain	domain	NOUN
ejde-761	190	12	,	,	PUNCT
ejde-761	190	13	the	the	DET
ejde-761	190	14	problem	problem	NOUN
ejde-761	190	15	in	in	ADP
ejde-761	190	16	(	(	PUNCT
ejde-761	190	17	1.1	1.1	NUM
ejde-761	190	18	)	)	PUNCT
ejde-761	190	19	is	be	AUX
ejde-761	190	20	then	then	ADV
ejde-761	190	21	reformulated	reformulate	VERB
ejde-761	190	22	as	as	ADP
ejde-761	190	23	−λγ′	−λγ′	PROPN
ejde-761	190	24	=	=	SYM
ejde-761	191	1	−(|γ′′|mγ′′)′′	−(|γ′′|mγ′′)′′	PROPN
ejde-761	191	2	+	+	NUM
ejde-761	191	3	|γ|p−1γ	|γ|p−1γ	NOUN
ejde-761	191	4	.	.	PUNCT
ejde-761	192	1	(	(	PUNCT
ejde-761	192	2	4.1	4.1	NUM
ejde-761	192	3	)	)	PUNCT
ejde-761	192	4	the	the	DET
ejde-761	192	5	next	next	ADJ
ejde-761	192	6	lemma	lemma	PROPN
ejde-761	192	7	aims	aim	VERB
ejde-761	192	8	at	at	ADP
ejde-761	192	9	characterizing	characterize	VERB
ejde-761	192	10	the	the	DET
ejde-761	192	11	tw	tw	NOUN
ejde-761	192	12	propagating	propagate	VERB
ejde-761	192	13	direction	direction	NOUN
ejde-761	192	14	.	.	PUNCT
ejde-761	193	1	this	this	DET
ejde-761	193	2	step	step	NOUN
ejde-761	193	3	is	be	AUX
ejde-761	193	4	relevant	relevant	ADJ
ejde-761	193	5	to	to	PART
ejde-761	193	6	properly	properly	ADV
ejde-761	193	7	describe	describe	VERB
ejde-761	193	8	the	the	DET
ejde-761	193	9	wave	wave	NOUN
ejde-761	193	10	dynamics	dynamic	NOUN
ejde-761	193	11	when	when	SCONJ
ejde-761	193	12	performing	perform	VERB
ejde-761	193	13	the	the	DET
ejde-761	193	14	numerical	numerical	ADJ
ejde-761	193	15	assessments	assessment	NOUN
ejde-761	193	16	.	.	PUNCT
ejde-761	194	1	lemma	lemma	PROPN
ejde-761	194	2	4.1	4.1	NUM
ejde-761	194	3	.	.	PUNCT
ejde-761	195	1	the	the	DET
ejde-761	195	2	tw	tw	PROPN
ejde-761	195	3	velocity	velocity	NOUN
ejde-761	195	4	λ	λ	PROPN
ejde-761	195	5	is	be	AUX
ejde-761	195	6	positive	positive	ADJ
ejde-761	195	7	,	,	PUNCT
ejde-761	195	8	equivalently	equivalently	ADV
ejde-761	195	9	,	,	PUNCT
ejde-761	195	10	the	the	DET
ejde-761	195	11	tw	tw	NOUN
ejde-761	195	12	motion	motion	NOUN
ejde-761	195	13	departs	depart	VERB
ejde-761	195	14	from	from	ADP
ejde-761	195	15	ξ	ξ	PROPN
ejde-761	195	16	→	→	SYM
ejde-761	195	17	−∞	−∞	NOUN
ejde-761	195	18	and	and	CCONJ
ejde-761	195	19	ends	end	VERB
ejde-761	195	20	in	in	ADP
ejde-761	195	21	ξ	ξ	PROPN
ejde-761	195	22	→	→	SYM
ejde-761	195	23	∞.	∞.	PROPN
ejde-761	195	24	proof	proof	NOUN
ejde-761	195	25	.	.	PUNCT
ejde-761	196	1	multiply	multiply	ADV
ejde-761	196	2	(	(	PUNCT
ejde-761	196	3	4.1	4.1	NUM
ejde-761	196	4	)	)	PUNCT
ejde-761	196	5	by	by	ADP
ejde-761	196	6	γ′	γ′	PROPN
ejde-761	196	7	:	:	PUNCT
ejde-761	196	8	,	,	PUNCT
ejde-761	196	9	−λ(γ′)2	−λ(γ′)2	PROPN
ejde-761	196	10	=	=	PUNCT
ejde-761	196	11	−(|γ′′|mγ′′)′′γ′	−(|γ′′|mγ′′)′′γ′	NOUN
ejde-761	196	12	+	+	CCONJ
ejde-761	196	13	|γ|p−1γγ′	|γ|p−1γγ′	NUM
ejde-761	196	14	,	,	PUNCT
ejde-761	196	15	(	(	PUNCT
ejde-761	196	16	4.2	4.2	NUM
ejde-761	196	17	)	)	PUNCT
ejde-761	196	18	and	and	CCONJ
ejde-761	196	19	consider	consider	VERB
ejde-761	196	20	the	the	DET
ejde-761	196	21	integration	integration	NOUN
ejde-761	196	22	from	from	ADP
ejde-761	196	23	−∞	−∞	X
ejde-761	196	24	to	to	ADP
ejde-761	196	25	∞.	∞.	PROPN
ejde-761	196	26	we	we	PRON
ejde-761	196	27	start	start	VERB
ejde-761	196	28	the	the	DET
ejde-761	196	29	evaluation	evaluation	NOUN
ejde-761	196	30	of	of	ADP
ejde-761	196	31	the	the	DET
ejde-761	196	32	integrals	integral	NOUN
ejde-761	196	33	by	by	ADP
ejde-761	196	34	the	the	DET
ejde-761	196	35	diffusive	diffusive	ADJ
ejde-761	196	36	term:∫	term:∫	NOUN
ejde-761	197	1	(	(	PUNCT
ejde-761	197	2	|γ′′|mγ′′)′′γ′	|γ′′|mγ′′)′′γ′	NUM
ejde-761	197	3	=	=	SYM
ejde-761	198	1	γ′(|γ′′|mγ′′)′	γ′(|γ′′|mγ′′)′	PROPN
ejde-761	199	1	−	−	PROPN
ejde-761	199	2	∫	∫	PROPN
ejde-761	199	3	(	(	PUNCT
ejde-761	199	4	|γ′′|mγ′′)′γ(2	|γ′′|mγ′′)′γ(2	PROPN
ejde-761	199	5	)	)	PUNCT
ejde-761	199	6	=	=	PUNCT
ejde-761	199	7	γ′(|γ′′|mγ′′)′	γ′(|γ′′|mγ′′)′	PROPN
ejde-761	199	8	−	−	PROPN
ejde-761	199	9	(	(	PUNCT
ejde-761	199	10	γ′(|γ′′|mγ′′)γ(2	γ′(|γ′′|mγ′′)γ(2	PROPN
ejde-761	199	11	)	)	PUNCT
ejde-761	199	12	−	−	PROPN
ejde-761	199	13	∫	∫	PROPN
ejde-761	199	14	γ′(|γ′′|mγ′′)γ(3	γ′(|γ′′|mγ′′)γ(3	PROPN
ejde-761	199	15	)	)	PUNCT
ejde-761	199	16	)	)	PUNCT
ejde-761	199	17	.	.	PUNCT
ejde-761	200	1	(	(	PUNCT
ejde-761	200	2	4.3	4.3	NUM
ejde-761	200	3	)	)	PUNCT
ejde-761	200	4	10	10	NUM
ejde-761	200	5	j.	j.	PROPN
ejde-761	200	6	l.	l.	PROPN
ejde-761	200	7	díaz	díaz	PROPN
ejde-761	200	8	palencia	palencia	PROPN
ejde-761	200	9	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	200	10	note	note	NOUN
ejde-761	200	11	that	that	SCONJ
ejde-761	200	12	the	the	DET
ejde-761	200	13	given	give	VERB
ejde-761	200	14	integrals	integral	NOUN
ejde-761	200	15	are	be	AUX
ejde-761	200	16	determined	determine	VERB
ejde-761	200	17	in	in	ADP
ejde-761	200	18	the	the	DET
ejde-761	200	19	limit	limit	NOUN
ejde-761	200	20	−∞	−∞	ADP
ejde-761	200	21	to	to	ADP
ejde-761	200	22	+	+	NOUN
ejde-761	200	23	∞	∞	PROPN
ejde-761	200	24	,	,	PUNCT
ejde-761	200	25	such	such	ADJ
ejde-761	200	26	that	that	SCONJ
ejde-761	200	27	the	the	DET
ejde-761	200	28	following	follow	VERB
ejde-761	200	29	asymptotic	asymptotic	ADJ
ejde-761	200	30	conditions	condition	NOUN
ejde-761	200	31	are	be	AUX
ejde-761	200	32	assumed	assume	VERB
ejde-761	200	33	to	to	PART
ejde-761	200	34	hold	hold	VERB
ejde-761	200	35	:	:	PUNCT
ejde-761	200	36	γ′(−∞	γ′(−∞	NUM
ejde-761	200	37	)	)	PUNCT
ejde-761	201	1	=	=	PUNCT
ejde-761	201	2	γ(2)(−∞	γ(2)(−∞	PROPN
ejde-761	201	3	)	)	PUNCT
ejde-761	201	4	=	=	SYM
ejde-761	201	5	γ(3)(−∞	γ(3)(−∞	PROPN
ejde-761	201	6	)	)	PUNCT
ejde-761	201	7	=	=	SYM
ejde-761	201	8	0	0	NUM
ejde-761	201	9	,	,	PUNCT
ejde-761	201	10	γ′(∞	γ′(∞	PROPN
ejde-761	201	11	)	)	PUNCT
ejde-761	201	12	=	=	SYM
ejde-761	202	1	γ(2)(∞	γ(2)(∞	PROPN
ejde-761	202	2	)	)	PUNCT
ejde-761	202	3	=	=	SYM
ejde-761	203	1	γ(3)(∞	γ(3)(∞	PROPN
ejde-761	203	2	)	)	PUNCT
ejde-761	203	3	=	=	SYM
ejde-761	203	4	0	0	X
ejde-761	203	5	.	.	PUNCT
ejde-761	204	1	(	(	PUNCT
ejde-761	204	2	4.4	4.4	NUM
ejde-761	204	3	)	)	PUNCT
ejde-761	204	4	consequently	consequently	ADV
ejde-761	204	5	,	,	PUNCT
ejde-761	204	6	∫	∫	PROPN
ejde-761	204	7	(	(	PUNCT
ejde-761	204	8	|γ′′|mγ′′)′′γ′	|γ′′|mγ′′)′′γ′	PROPN
ejde-761	204	9	=	=	SYM
ejde-761	204	10	0	0	PROPN
ejde-761	204	11	.	.	PUNCT
ejde-761	205	1	(	(	PUNCT
ejde-761	205	2	4.5	4.5	NUM
ejde-761	205	3	)	)	PUNCT
ejde-761	205	4	now	now	ADV
ejde-761	205	5	,	,	PUNCT
ejde-761	205	6	for	for	ADP
ejde-761	205	7	the	the	DET
ejde-761	205	8	term	term	NOUN
ejde-761	205	9	|γ|p−1γγ′	|γ|p−1γγ′	NOUN
ejde-761	205	10	,	,	PUNCT
ejde-761	205	11	we	we	PRON
ejde-761	205	12	shall	shall	AUX
ejde-761	205	13	considered	consider	VERB
ejde-761	205	14	that	that	SCONJ
ejde-761	205	15	there	there	PRON
ejde-761	205	16	exists	exist	VERB
ejde-761	205	17	a	a	DET
ejde-761	205	18	jump	jump	NOUN
ejde-761	205	19	of	of	ADP
ejde-761	205	20	magnitude	magnitude	NOUN
ejde-761	205	21	c	c	NOUN
ejde-761	205	22	in	in	ADP
ejde-761	205	23	the	the	DET
ejde-761	205	24	tw	tw	NOUN
ejde-761	205	25	profile	profile	NOUN
ejde-761	205	26	.	.	PUNCT
ejde-761	206	1	this	this	DET
ejde-761	206	2	fact	fact	NOUN
ejde-761	206	3	will	will	AUX
ejde-761	206	4	be	be	AUX
ejde-761	206	5	further	far	ADV
ejde-761	206	6	developed	develop	VERB
ejde-761	206	7	in	in	ADP
ejde-761	206	8	the	the	DET
ejde-761	206	9	numerical	numerical	ADJ
ejde-761	206	10	exercise	exercise	NOUN
ejde-761	206	11	,	,	PUNCT
ejde-761	206	12	but	but	CCONJ
ejde-761	206	13	as	as	ADP
ejde-761	206	14	a	a	DET
ejde-761	206	15	first	first	ADJ
ejde-761	206	16	description	description	NOUN
ejde-761	206	17	,	,	PUNCT
ejde-761	206	18	it	it	PRON
ejde-761	206	19	shall	shall	AUX
ejde-761	206	20	be	be	AUX
ejde-761	206	21	noted	note	VERB
ejde-761	206	22	that	that	SCONJ
ejde-761	206	23	such	such	ADJ
ejde-761	206	24	jump	jump	NOUN
ejde-761	206	25	represents	represent	VERB
ejde-761	206	26	a	a	DET
ejde-761	206	27	transition	transition	NOUN
ejde-761	206	28	from	from	ADP
ejde-761	206	29	a	a	DET
ejde-761	206	30	finite	finite	ADJ
ejde-761	206	31	mass	mass	NOUN
ejde-761	206	32	initial	initial	ADJ
ejde-761	206	33	condition	condition	NOUN
ejde-761	206	34	to	to	ADP
ejde-761	206	35	a	a	DET
ejde-761	206	36	null	null	ADJ
ejde-761	206	37	condition	condition	NOUN
ejde-761	206	38	(	(	PUNCT
ejde-761	206	39	at	at	ADP
ejde-761	206	40	infinity	infinity	NOUN
ejde-761	206	41	)	)	PUNCT
ejde-761	206	42	where	where	SCONJ
ejde-761	206	43	the	the	DET
ejde-761	206	44	instabilities	instability	NOUN
ejde-761	206	45	are	be	AUX
ejde-761	206	46	characterized	characterize	VERB
ejde-761	206	47	.	.	PUNCT
ejde-761	207	1	then∫	then∫	NOUN
ejde-761	207	2	|γ|p−1γγ′	|γ|p−1γγ′	NOUN
ejde-761	208	1	=	=	PUNCT
ejde-761	208	2	γpγ	γpγ	ADJ
ejde-761	208	3	−	−	PROPN
ejde-761	209	1	p	p	NOUN
ejde-761	209	2	∫	∫	PROPN
ejde-761	209	3	γp	γp	X
ejde-761	209	4	=	=	PUNCT
ejde-761	209	5	γpγ	γpγ	ADJ
ejde-761	210	1	−	−	PROPN
ejde-761	210	2	p	p	NOUN
ejde-761	210	3	∫	∫	PROPN
ejde-761	210	4	γp−2γ2	γp−2γ2	NOUN
ejde-761	210	5	=	=	PUNCT
ejde-761	210	6	γpγ	γpγ	ADJ
ejde-761	210	7	−	−	NOUN
ejde-761	210	8	p	p	NOUN
ejde-761	210	9	p−	p−	NOUN
ejde-761	210	10	1	1	NUM
ejde-761	210	11	γp+1	γp+1	NOUN
ejde-761	210	12	+	+	NOUN
ejde-761	210	13	2p	2p	NUM
ejde-761	210	14	p−	p−	NOUN
ejde-761	210	15	1	1	NUM
ejde-761	210	16	∫	∫	NOUN
ejde-761	210	17	γpγ′.	γpγ′.	PROPN
ejde-761	210	18	(	(	PUNCT
ejde-761	210	19	4.6	4.6	NUM
ejde-761	210	20	)	)	PUNCT
ejde-761	210	21	consequently	consequently	ADV
ejde-761	210	22	,	,	PUNCT
ejde-761	210	23	based	base	VERB
ejde-761	210	24	on	on	ADP
ejde-761	210	25	the	the	DET
ejde-761	210	26	consideration	consideration	NOUN
ejde-761	210	27	of	of	ADP
ejde-761	210	28	the	the	DET
ejde-761	210	29	mentioned	mention	VERB
ejde-761	210	30	jump	jump	NOUN
ejde-761	210	31	of	of	ADP
ejde-761	210	32	magnitude	magnitude	NOUN
ejde-761	210	33	c	c	X
ejde-761	210	34	,	,	PUNCT
ejde-761	210	35	we	we	PRON
ejde-761	210	36	have	have	VERB
ejde-761	210	37	∫	∫	PROPN
ejde-761	211	1	|γ|p−1γγ′	|γ|p−1γγ′	NOUN
ejde-761	211	2	=	=	PUNCT
ejde-761	211	3	γpγ	γpγ	ADJ
ejde-761	211	4	−	−	PROPN
ejde-761	211	5	p	p	NOUN
ejde-761	211	6	p−1γ	p−1γ	NOUN
ejde-761	211	7	p+1	p+1	NOUN
ejde-761	211	8	1−	1−	NUM
ejde-761	211	9	2p	2p	NUM
ejde-761	211	10	p−1	p−1	PROPN
ejde-761	211	11	=	=	PUNCT
ejde-761	211	12	γp+1	γp+1	PROPN
ejde-761	211	13	p+	p+	VERB
ejde-761	211	14	1	1	NUM
ejde-761	211	15	=	=	SYM
ejde-761	211	16	1	1	NUM
ejde-761	211	17	p+	p+	NOUN
ejde-761	211	18	1	1	NUM
ejde-761	211	19	(	(	PUNCT
ejde-761	211	20	γp+1(∞)−	γp+1(∞)−	X
ejde-761	211	21	γp+1(−∞	γp+1(−∞	PROPN
ejde-761	211	22	)	)	PUNCT
ejde-761	211	23	)	)	PUNCT
ejde-761	212	1	=	=	SYM
ejde-761	212	2	1	1	NUM
ejde-761	212	3	p+	p+	NOUN
ejde-761	212	4	1	1	NUM
ejde-761	212	5	(	(	PUNCT
ejde-761	212	6	0−	0−	NUM
ejde-761	212	7	cp+1	cp+1	NUM
ejde-761	212	8	)	)	PUNCT
ejde-761	212	9	.	.	PUNCT
ejde-761	213	1	(	(	PUNCT
ejde-761	213	2	4.7	4.7	NUM
ejde-761	213	3	)	)	PUNCT
ejde-761	213	4	finally	finally	ADV
ejde-761	213	5	and	and	CCONJ
ejde-761	213	6	upon	upon	SCONJ
ejde-761	213	7	recovery	recovery	NOUN
ejde-761	213	8	of	of	ADP
ejde-761	213	9	the	the	DET
ejde-761	213	10	expression	expression	NOUN
ejde-761	213	11	(	(	PUNCT
ejde-761	213	12	4.2	4.2	NUM
ejde-761	213	13	)	)	PUNCT
ejde-761	213	14	,	,	PUNCT
ejde-761	213	15	the	the	DET
ejde-761	213	16	following	follow	VERB
ejde-761	213	17	holds	hold	NOUN
ejde-761	213	18	,	,	PUNCT
ejde-761	213	19	−λ	−λ	PROPN
ejde-761	213	20	∫	∫	PROPN
ejde-761	213	21	(	(	PUNCT
ejde-761	213	22	γ′)2	γ′)2	NOUN
ejde-761	213	23	=	=	SYM
ejde-761	213	24	0−	0−	NUM
ejde-761	213	25	1	1	NUM
ejde-761	213	26	p+	p+	NOUN
ejde-761	213	27	1	1	NUM
ejde-761	213	28	cp+1	cp+1	NOUN
ejde-761	213	29	,	,	PUNCT
ejde-761	213	30	(	(	PUNCT
ejde-761	213	31	4.8	4.8	NUM
ejde-761	213	32	)	)	PUNCT
ejde-761	213	33	which	which	PRON
ejde-761	213	34	leads	lead	VERB
ejde-761	213	35	to	to	ADP
ejde-761	213	36	λ	λ	PROPN
ejde-761	213	37	=	=	SYM
ejde-761	213	38	1	1	NUM
ejde-761	213	39	p+	p+	NOUN
ejde-761	213	40	1	1	NUM
ejde-761	213	41	cp+1∫	cp+1∫	NOUN
ejde-761	213	42	(	(	PUNCT
ejde-761	213	43	γ′)2	γ′)2	NOUN
ejde-761	213	44	.	.	PUNCT
ejde-761	214	1	(	(	PUNCT
ejde-761	214	2	4.9	4.9	NUM
ejde-761	214	3	)	)	PUNCT
ejde-761	214	4	this	this	DET
ejde-761	214	5	last	last	ADJ
ejde-761	214	6	expression	expression	NOUN
ejde-761	214	7	permits	permit	VERB
ejde-761	214	8	us	we	PRON
ejde-761	214	9	to	to	PART
ejde-761	214	10	conclude	conclude	VERB
ejde-761	214	11	that	that	PRON
ejde-761	214	12	λ	λ	PROPN
ejde-761	214	13	>	>	X
ejde-761	214	14	0	0	PUNCT
ejde-761	214	15	as	as	SCONJ
ejde-761	214	16	claimed	claim	VERB
ejde-761	214	17	.	.	PUNCT
ejde-761	215	1	□	□	PUNCT
ejde-761	215	2	4.1	4.1	NUM
ejde-761	215	3	.	.	PUNCT
ejde-761	215	4	travelling	travel	VERB
ejde-761	215	5	wave	wave	NOUN
ejde-761	215	6	instabilities	instability	NOUN
ejde-761	215	7	.	.	PUNCT
ejde-761	216	1	the	the	DET
ejde-761	216	2	tws	tws	PROPN
ejde-761	216	3	formulation	formulation	NOUN
ejde-761	216	4	in	in	ADP
ejde-761	216	5	expression	expression	NOUN
ejde-761	216	6	(	(	PUNCT
ejde-761	216	7	4.1	4.1	NUM
ejde-761	216	8	)	)	PUNCT
ejde-761	216	9	has	have	VERB
ejde-761	216	10	one	one	NUM
ejde-761	216	11	critical	critical	ADJ
ejde-761	216	12	solution	solution	NOUN
ejde-761	216	13	at	at	ADP
ejde-761	216	14	γ	γ	X
ejde-761	216	15	=	=	SYM
ejde-761	216	16	0	0	NUM
ejde-761	216	17	.	.	PUNCT
ejde-761	217	1	the	the	DET
ejde-761	217	2	aim	aim	NOUN
ejde-761	217	3	of	of	ADP
ejde-761	217	4	this	this	DET
ejde-761	217	5	subsection	subsection	NOUN
ejde-761	217	6	is	be	AUX
ejde-761	217	7	to	to	PART
ejde-761	217	8	analyze	analyze	VERB
ejde-761	217	9	the	the	DET
ejde-761	217	10	oscillating	oscillate	VERB
ejde-761	217	11	properties	property	NOUN
ejde-761	217	12	of	of	ADP
ejde-761	217	13	any	any	DET
ejde-761	217	14	solution	solution	NOUN
ejde-761	217	15	in	in	ADP
ejde-761	217	16	the	the	DET
ejde-761	217	17	proximity	proximity	NOUN
ejde-761	217	18	of	of	ADP
ejde-761	217	19	the	the	DET
ejde-761	217	20	null	null	ADJ
ejde-761	217	21	solution	solution	NOUN
ejde-761	217	22	.	.	PUNCT
ejde-761	218	1	the	the	DET
ejde-761	218	2	oscillations	oscillation	NOUN
ejde-761	218	3	induced	induce	VERB
ejde-761	218	4	by	by	ADP
ejde-761	218	5	the	the	DET
ejde-761	218	6	higher	high	ADJ
ejde-761	218	7	order	order	NOUN
ejde-761	218	8	non	non	ADJ
ejde-761	218	9	-	-	ADJ
ejde-761	218	10	linear	linear	ADJ
ejde-761	218	11	diffusion	diffusion	NOUN
ejde-761	218	12	are	be	AUX
ejde-761	218	13	studied	study	VERB
ejde-761	218	14	with	with	ADP
ejde-761	218	15	the	the	DET
ejde-761	218	16	introduced	introduce	VERB
ejde-761	218	17	norm	norm	NOUN
ejde-761	218	18	in	in	ADP
ejde-761	218	19	(	(	PUNCT
ejde-761	218	20	2.3	2.3	NUM
ejde-761	218	21	)	)	PUNCT
ejde-761	218	22	.	.	PUNCT
ejde-761	219	1	afterward	afterward	ADV
ejde-761	219	2	,	,	PUNCT
ejde-761	219	3	this	this	DET
ejde-761	219	4	chapter	chapter	NOUN
ejde-761	219	5	aims	aim	VERB
ejde-761	219	6	at	at	ADP
ejde-761	219	7	showing	show	VERB
ejde-761	219	8	that	that	SCONJ
ejde-761	219	9	the	the	DET
ejde-761	219	10	null	null	ADJ
ejde-761	219	11	solution	solution	NOUN
ejde-761	219	12	acts	act	VERB
ejde-761	219	13	as	as	ADP
ejde-761	219	14	an	an	DET
ejde-761	219	15	attractor	attractor	NOUN
ejde-761	219	16	of	of	ADP
ejde-761	219	17	solutions	solution	NOUN
ejde-761	219	18	hindering	hinder	VERB
ejde-761	219	19	the	the	DET
ejde-761	219	20	possible	possible	ADJ
ejde-761	219	21	nucleation	nucleation	NOUN
ejde-761	219	22	of	of	ADP
ejde-761	219	23	blow	blow	NOUN
ejde-761	219	24	-	-	PUNCT
ejde-761	219	25	up	up	ADP
ejde-761	219	26	profiles	profile	NOUN
ejde-761	219	27	.	.	PUNCT
ejde-761	220	1	the	the	DET
ejde-761	220	2	study	study	NOUN
ejde-761	220	3	of	of	ADP
ejde-761	220	4	oscillations	oscillation	NOUN
ejde-761	220	5	close	close	ADJ
ejde-761	220	6	to	to	ADP
ejde-761	220	7	the	the	DET
ejde-761	220	8	null	null	ADJ
ejde-761	220	9	solution	solution	NOUN
ejde-761	220	10	follows	follow	VERB
ejde-761	220	11	from	from	ADP
ejde-761	220	12	a	a	DET
ejde-761	220	13	theorem	theorem	NOUN
ejde-761	220	14	introduced	introduce	VERB
ejde-761	220	15	to	to	PART
ejde-761	220	16	study	study	VERB
ejde-761	220	17	the	the	DET
ejde-761	220	18	kuramoto	kuramoto	NOUN
ejde-761	220	19	-	-	PUNCT
ejde-761	220	20	sivashinsky	sivashinsky	NOUN
ejde-761	220	21	equation	equation	NOUN
ejde-761	220	22	(	(	PUNCT
ejde-761	220	23	see	see	VERB
ejde-761	220	24	[	[	X
ejde-761	220	25	43	43	NUM
ejde-761	220	26	]	]	PUNCT
ejde-761	220	27	and	and	CCONJ
ejde-761	220	28	references	reference	NOUN
ejde-761	220	29	therein	therein	ADV
ejde-761	220	30	)	)	PUNCT
ejde-761	220	31	along	along	ADP
ejde-761	220	32	with	with	ADP
ejde-761	220	33	other	other	ADJ
ejde-761	220	34	equations	equation	NOUN
ejde-761	220	35	,	,	PUNCT
ejde-761	220	36	particularly	particularly	ADV
ejde-761	220	37	the	the	DET
ejde-761	220	38	cahn	cahn	NOUN
ejde-761	220	39	-	-	PUNCT
ejde-761	220	40	hilliard	hilliard	NOUN
ejde-761	220	41	equation	equation	NOUN
ejde-761	220	42	(	(	PUNCT
ejde-761	220	43	see	see	VERB
ejde-761	220	44	[	[	X
ejde-761	220	45	28	28	NUM
ejde-761	220	46	]	]	PUNCT
ejde-761	220	47	)	)	PUNCT
ejde-761	220	48	and	and	CCONJ
ejde-761	220	49	a	a	DET
ejde-761	220	50	sixth	sixth	ADJ
ejde-761	220	51	order	order	NOUN
ejde-761	220	52	diffusion	diffusion	NOUN
ejde-761	220	53	equation	equation	NOUN
ejde-761	220	54	(	(	PUNCT
ejde-761	220	55	see	see	VERB
ejde-761	220	56	[	[	X
ejde-761	220	57	34	34	NUM
ejde-761	220	58	]	]	PUNCT
ejde-761	220	59	)	)	PUNCT
ejde-761	220	60	.	.	PUNCT
ejde-761	221	1	nonetheless	nonetheless	ADV
ejde-761	221	2	,	,	PUNCT
ejde-761	221	3	for	for	ADP
ejde-761	221	4	our	our	PRON
ejde-761	221	5	present	present	ADJ
ejde-761	221	6	case	case	NOUN
ejde-761	221	7	,	,	PUNCT
ejde-761	221	8	the	the	DET
ejde-761	221	9	higher	high	ADJ
ejde-761	221	10	order	order	NOUN
ejde-761	221	11	p	p	X
ejde-761	221	12	-	-	PUNCT
ejde-761	221	13	laplacian	laplacian	ADJ
ejde-761	221	14	operator	operator	NOUN
ejde-761	221	15	induces	induce	VERB
ejde-761	221	16	a	a	DET
ejde-761	221	17	set	set	NOUN
ejde-761	221	18	of	of	ADP
ejde-761	221	19	changes	change	NOUN
ejde-761	221	20	in	in	ADP
ejde-761	221	21	the	the	DET
ejde-761	221	22	mentioned	mention	VERB
ejde-761	221	23	theorem	theorem	NOUN
ejde-761	221	24	for	for	ADP
ejde-761	221	25	the	the	DET
ejde-761	221	26	cited	cite	VERB
ejde-761	221	27	equations	equation	NOUN
ejde-761	221	28	.	.	PUNCT
ejde-761	222	1	to	to	ADP
ejde-761	222	2	this	this	DET
ejde-761	222	3	end	end	NOUN
ejde-761	222	4	,	,	PUNCT
ejde-761	222	5	the	the	DET
ejde-761	222	6	theorem	theorem	NOUN
ejde-761	222	7	is	be	AUX
ejde-761	222	8	divided	divide	VERB
ejde-761	222	9	into	into	ADP
ejde-761	222	10	four	four	NUM
ejde-761	222	11	lemmas	lemmas	ADJ
ejde-761	222	12	,	,	PUNCT
ejde-761	222	13	so	so	SCONJ
ejde-761	222	14	that	that	SCONJ
ejde-761	222	15	a	a	DET
ejde-761	222	16	instability	instability	NOUN
ejde-761	222	17	statement	statement	NOUN
ejde-761	222	18	is	be	AUX
ejde-761	222	19	proved	prove	VERB
ejde-761	222	20	.	.	PUNCT
ejde-761	223	1	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	223	2	instability	instability	NOUN
ejde-761	223	3	of	of	ADP
ejde-761	223	4	energy	energy	NOUN
ejde-761	223	5	solutions	solution	NOUN
ejde-761	223	6	11	11	NUM
ejde-761	223	7	the	the	DET
ejde-761	223	8	fist	fist	NOUN
ejde-761	223	9	lemma	lemma	PROPN
ejde-761	223	10	introduces	introduce	VERB
ejde-761	223	11	the	the	DET
ejde-761	223	12	principle	principle	NOUN
ejde-761	223	13	of	of	ADP
ejde-761	223	14	oscillations	oscillation	NOUN
ejde-761	223	15	(	(	PUNCT
ejde-761	223	16	or	or	CCONJ
ejde-761	223	17	instabilities	instability	NOUN
ejde-761	223	18	)	)	PUNCT
ejde-761	223	19	for	for	ADP
ejde-761	223	20	any	any	DET
ejde-761	223	21	energy	energy	NOUN
ejde-761	223	22	solution	solution	NOUN
ejde-761	223	23	.	.	PUNCT
ejde-761	224	1	this	this	PRON
ejde-761	224	2	is	be	AUX
ejde-761	224	3	shown	show	VERB
ejde-761	224	4	based	base	VERB
ejde-761	224	5	on	on	ADP
ejde-761	224	6	the	the	DET
ejde-761	224	7	already	already	ADV
ejde-761	224	8	introduced	introduce	VERB
ejde-761	224	9	norms	norm	NOUN
ejde-761	224	10	in	in	ADP
ejde-761	224	11	(	(	PUNCT
ejde-761	224	12	2.3	2.3	NUM
ejde-761	224	13	)	)	PUNCT
ejde-761	224	14	and	and	CCONJ
ejde-761	224	15	(	(	PUNCT
ejde-761	224	16	2.7	2.7	NUM
ejde-761	224	17	)	)	PUNCT
ejde-761	224	18	.	.	PUNCT
ejde-761	225	1	lemma	lemma	PROPN
ejde-761	225	2	4.2	4.2	NUM
ejde-761	225	3	.	.	PUNCT
ejde-761	226	1	each	each	DET
ejde-761	226	2	oscillating	oscillate	VERB
ejde-761	226	3	energy	energy	NOUN
ejde-761	226	4	solution	solution	NOUN
ejde-761	226	5	u(x	u(x	NOUN
ejde-761	226	6	,	,	PUNCT
ejde-761	226	7	t	t	NOUN
ejde-761	226	8	)	)	PUNCT
ejde-761	226	9	∈	∈	PROPN
ejde-761	226	10	l2(rn	l2(rn	PROPN
ejde-761	226	11	)	)	PUNCT
ejde-761	226	12	is	be	AUX
ejde-761	226	13	bounded	bound	VERB
ejde-761	226	14	by	by	ADP
ejde-761	226	15	the	the	DET
ejde-761	226	16	sobolev	sobolev	NOUN
ejde-761	226	17	norms	norm	NOUN
ejde-761	226	18	(	(	PUNCT
ejde-761	226	19	2.3	2.3	NUM
ejde-761	226	20	)	)	PUNCT
ejde-761	226	21	and	and	CCONJ
ejde-761	226	22	(	(	PUNCT
ejde-761	226	23	2.7	2.7	NUM
ejde-761	226	24	)	)	PUNCT
ejde-761	226	25	i.e.	i.e.	X
ejde-761	226	26	∥u∥l2	∥u∥l2	X
ejde-761	226	27	≤	≤	NUM
ejde-761	226	28	k1∥u∥h4	k1∥u∥h4	NOUN
ejde-761	226	29	,	,	PUNCT
ejde-761	226	30	∥u∥l2	∥u∥l2	PROPN
ejde-761	226	31	≤	≤	PROPN
ejde-761	226	32	k2∥u∥θ	k2∥u∥θ	PROPN
ejde-761	226	33	.	.	PUNCT
ejde-761	227	1	(	(	PUNCT
ejde-761	227	2	4.10	4.10	NUM
ejde-761	227	3	)	)	PUNCT
ejde-761	227	4	proof	proof	NOUN
ejde-761	227	5	.	.	PUNCT
ejde-761	228	1	the	the	DET
ejde-761	228	2	first	first	ADJ
ejde-761	228	3	condition	condition	NOUN
ejde-761	228	4	is	be	AUX
ejde-761	228	5	trivially	trivially	ADV
ejde-761	228	6	shown	show	VERB
ejde-761	228	7	in	in	ADP
ejde-761	228	8	virtue	virtue	NOUN
ejde-761	228	9	of	of	ADP
ejde-761	228	10	the	the	DET
ejde-761	228	11	norm	norm	NOUN
ejde-761	228	12	h4	h4	NOUN
ejde-761	228	13	defined	define	VERB
ejde-761	228	14	in	in	ADP
ejde-761	228	15	(	(	PUNCT
ejde-761	228	16	2.7	2.7	NUM
ejde-761	228	17	)	)	PUNCT
ejde-761	228	18	,	,	PUNCT
ejde-761	228	19	∥u∥h4	∥u∥h4	NOUN
ejde-761	228	20	=	=	SYM
ejde-761	228	21	∥u∥l2	∥u∥l2	ADJ
ejde-761	228	22	+	+	SYM
ejde-761	228	23	∥∇4u∥l2	∥∇4u∥l2	ADJ
ejde-761	228	24	≥	≥	NOUN
ejde-761	228	25	∥u∥l2	∥u∥l2	NOUN
ejde-761	228	26	.	.	PUNCT
ejde-761	229	1	(	(	PUNCT
ejde-761	229	2	4.11	4.11	NUM
ejde-761	229	3	)	)	PUNCT
ejde-761	229	4	given	give	VERB
ejde-761	229	5	the	the	DET
ejde-761	229	6	positivity	positivity	NOUN
ejde-761	229	7	of	of	ADP
ejde-761	229	8	any	any	DET
ejde-761	229	9	norm	norm	NOUN
ejde-761	229	10	,	,	PUNCT
ejde-761	229	11	this	this	DET
ejde-761	229	12	last	last	ADJ
ejde-761	229	13	expression	expression	NOUN
ejde-761	229	14	concludes	conclude	VERB
ejde-761	229	15	on	on	ADP
ejde-761	229	16	∥u∥l2	∥u∥l2	ADJ
ejde-761	229	17	≤	≤	NOUN
ejde-761	229	18	∥u∥h4	∥u∥h4	NOUN
ejde-761	229	19	,	,	PUNCT
ejde-761	229	20	i.e.	i.e.	X
ejde-761	229	21	k1	k1	X
ejde-761	229	22	=	=	SYM
ejde-761	229	23	1	1	X
ejde-761	229	24	.	.	PUNCT
ejde-761	230	1	the	the	DET
ejde-761	230	2	next	next	ADJ
ejde-761	230	3	inequality	inequality	NOUN
ejde-761	230	4	is	be	AUX
ejde-761	230	5	shown	show	VERB
ejde-761	230	6	considering	consider	VERB
ejde-761	230	7	the	the	DET
ejde-761	230	8	expression	expression	NOUN
ejde-761	230	9	(	(	PUNCT
ejde-761	230	10	2.3	2.3	NUM
ejde-761	230	11	):	):	PUNCT
ejde-761	230	12	∥u∥2l2	∥u∥2l2	PROPN
ejde-761	230	13	≤	≤	NUM
ejde-761	230	14	∫	∫	PROPN
ejde-761	230	15	rn	rn	PROPN
ejde-761	230	16	4∑	4∑	PROPN
ejde-761	230	17	j=0	j=0	VERB
ejde-761	230	18	|dju(z)|2dz	|dju(z)|2dz	PROPN
ejde-761	230	19	≤	≤	PROPN
ejde-761	230	20	∫	∫	PROPN
ejde-761	230	21	rn	rn	PROPN
ejde-761	230	22	θ(z	θ(z	PROPN
ejde-761	230	23	)	)	PUNCT
ejde-761	230	24	4∑	4∑	NOUN
ejde-761	230	25	j=0	j=0	VERB
ejde-761	230	26	|dju(z)|2dz	|dju(z)|2dz	NOUN
ejde-761	230	27	=	=	SYM
ejde-761	230	28	∥u∥2θ	∥u∥2θ	NOUN
ejde-761	230	29	,	,	PUNCT
ejde-761	230	30	(	(	PUNCT
ejde-761	230	31	4.12	4.12	NUM
ejde-761	230	32	)	)	PUNCT
ejde-761	230	33	a.e	a.e	PROPN
ejde-761	230	34	.	.	PROPN
ejde-761	231	1	in	in	ADP
ejde-761	231	2	rn	rn	PROPN
ejde-761	231	3	.	.	PUNCT
ejde-761	232	1	then	then	ADV
ejde-761	232	2	,	,	PUNCT
ejde-761	232	3	it	it	PRON
ejde-761	232	4	suffices	suffice	VERB
ejde-761	232	5	to	to	PART
ejde-761	232	6	consider	consider	VERB
ejde-761	232	7	k2	k2	NOUN
ejde-761	232	8	=	=	NOUN
ejde-761	232	9	1	1	NUM
ejde-761	232	10	.	.	X
ejde-761	232	11	□	□	PUNCT
ejde-761	232	12	now	now	ADV
ejde-761	232	13	,	,	PUNCT
ejde-761	232	14	to	to	PART
ejde-761	232	15	introduce	introduce	VERB
ejde-761	232	16	the	the	DET
ejde-761	232	17	tw	tw	NOUN
ejde-761	232	18	convergence	convergence	NOUN
ejde-761	232	19	analysis	analysis	NOUN
ejde-761	232	20	,	,	PUNCT
ejde-761	232	21	we	we	PRON
ejde-761	232	22	introduce	introduce	VERB
ejde-761	232	23	the	the	DET
ejde-761	232	24	function	function	NOUN
ejde-761	232	25	w(x	w(x	PROPN
ejde-761	232	26	,	,	PUNCT
ejde-761	232	27	t	t	PROPN
ejde-761	232	28	)	)	PUNCT
ejde-761	232	29	=	=	SYM
ejde-761	232	30	u(x	u(x	PROPN
ejde-761	232	31	,	,	PUNCT
ejde-761	232	32	t)−	t)−	PROPN
ejde-761	232	33	φ(x	φ(x	PROPN
ejde-761	232	34	,	,	PUNCT
ejde-761	232	35	t	t	PROPN
ejde-761	232	36	)	)	PUNCT
ejde-761	232	37	,	,	PUNCT
ejde-761	232	38	where	where	SCONJ
ejde-761	232	39	φ(x	φ(x	PROPN
ejde-761	232	40	,	,	PUNCT
ejde-761	232	41	t	t	PROPN
ejde-761	232	42	)	)	PUNCT
ejde-761	232	43	represents	represent	VERB
ejde-761	232	44	a	a	DET
ejde-761	232	45	perturbation	perturbation	NOUN
ejde-761	232	46	randomly	randomly	ADV
ejde-761	232	47	small	small	ADJ
ejde-761	232	48	so	so	SCONJ
ejde-761	232	49	as	as	SCONJ
ejde-761	232	50	to	to	PART
ejde-761	232	51	ensure	ensure	VERB
ejde-761	232	52	the	the	DET
ejde-761	232	53	tw	tw	NOUN
ejde-761	232	54	profiles	profile	NOUN
ejde-761	232	55	convergence	convergence	NOUN
ejde-761	232	56	.	.	PUNCT
ejde-761	233	1	particularly	particularly	ADV
ejde-761	233	2	and	and	CCONJ
ejde-761	233	3	close	close	ADJ
ejde-761	233	4	to	to	ADP
ejde-761	233	5	the	the	DET
ejde-761	233	6	null	null	ADJ
ejde-761	233	7	solution	solution	NOUN
ejde-761	233	8	,	,	PUNCT
ejde-761	233	9	φ(x	φ(x	PROPN
ejde-761	233	10	,	,	PUNCT
ejde-761	233	11	t	t	PROPN
ejde-761	233	12	)	)	PUNCT
ejde-761	233	13	is	be	AUX
ejde-761	233	14	requested	request	VERB
ejde-761	233	15	to	to	PART
ejde-761	233	16	satisfy	satisfy	VERB
ejde-761	233	17	(	(	PUNCT
ejde-761	233	18	see	see	VERB
ejde-761	233	19	lemma	lemma	PROPN
ejde-761	233	20	3.1	3.1	NUM
ejde-761	233	21	along	along	ADP
ejde-761	233	22	with	with	ADP
ejde-761	233	23	inequality	inequality	NOUN
ejde-761	233	24	(	(	PUNCT
ejde-761	233	25	3.8	3.8	NUM
ejde-761	233	26	)	)	PUNCT
ejde-761	233	27	):	):	PUNCT
ejde-761	233	28	∥φ∥l2	∥φ∥l2	NOUN
ejde-761	233	29	≤	≤	NUM
ejde-761	233	30	∥φ∥θ	∥φ∥θ	VERB
ejde-761	233	31	≤	≤	NOUN
ejde-761	233	32	σ∥φ∥hm	σ∥φ∥hm	CCONJ
ejde-761	233	33	ρ	ρ	NOUN
ejde-761	233	34	.	.	PUNCT
ejde-761	234	1	(	(	PUNCT
ejde-761	234	2	4.13	4.13	NUM
ejde-761	234	3	)	)	PUNCT
ejde-761	234	4	convergence	convergence	NOUN
ejde-761	234	5	requires	require	VERB
ejde-761	234	6	σ	σ	PROPN
ejde-761	234	7	→	→	SYM
ejde-761	234	8	0	0	NUM
ejde-761	234	9	,	,	PUNCT
ejde-761	234	10	i.e.	i.e.	X
ejde-761	234	11	a	a	DET
ejde-761	234	12	mollification	mollification	NOUN
ejde-761	234	13	on	on	ADP
ejde-761	234	14	the	the	DET
ejde-761	234	15	function	function	NOUN
ejde-761	234	16	φ	φ	NOUN
ejde-761	234	17	.	.	PUNCT
ejde-761	235	1	now	now	ADV
ejde-761	235	2	,	,	PUNCT
ejde-761	235	3	the	the	DET
ejde-761	235	4	problem	problem	NOUN
ejde-761	235	5	(	(	PUNCT
ejde-761	235	6	1.1	1.1	NUM
ejde-761	235	7	)	)	PUNCT
ejde-761	235	8	is	be	AUX
ejde-761	235	9	hence	hence	ADV
ejde-761	235	10	formulated	formulate	VERB
ejde-761	235	11	in	in	ADP
ejde-761	235	12	terms	term	NOUN
ejde-761	235	13	of	of	ADP
ejde-761	235	14	w(x	w(x	PROPN
ejde-761	235	15	,	,	PUNCT
ejde-761	235	16	t	t	PROPN
ejde-761	235	17	)	)	PUNCT
ejde-761	235	18	and	and	CCONJ
ejde-761	235	19	φ(x	φ(x	PROPN
ejde-761	235	20	,	,	PUNCT
ejde-761	235	21	t	t	PROPN
ejde-761	235	22	)	)	PUNCT
ejde-761	235	23	as	as	ADP
ejde-761	235	24	wt	wt	PROPN
ejde-761	235	25	+	+	NOUN
ejde-761	235	26	φt	φt	NOUN
ejde-761	235	27	=	=	SYM
ejde-761	235	28	−∆	−∆	NOUN
ejde-761	235	29	(	(	PUNCT
ejde-761	235	30	∞∑	∞∑	NUM
ejde-761	235	31	k=0	k=0	PROPN
ejde-761	235	32	(	(	PUNCT
ejde-761	235	33	m	m	VERB
ejde-761	235	34	k	k	NOUN
ejde-761	235	35	)	)	PUNCT
ejde-761	235	36	(	(	PUNCT
ejde-761	235	37	∆w)m−k(∆φ)k	∆w)m−k(∆φ)k	PROPN
ejde-761	235	38	∆(w	∆(w	PROPN
ejde-761	235	39	+	+	CCONJ
ejde-761	235	40	φ	φ	NUM
ejde-761	235	41	)	)	PUNCT
ejde-761	235	42	)	)	PUNCT
ejde-761	236	1	+	+	CCONJ
ejde-761	236	2	∞∑	∞∑	NUM
ejde-761	236	3	j=0	j=0	PROPN
ejde-761	236	4	(	(	PUNCT
ejde-761	236	5	p	p	PROPN
ejde-761	236	6	j	j	PROPN
ejde-761	236	7	)	)	PUNCT
ejde-761	236	8	wp−jφj	wp−jφj	X
ejde-761	236	9	.	.	PUNCT
ejde-761	237	1	(	(	PUNCT
ejde-761	237	2	4.14	4.14	NUM
ejde-761	237	3	)	)	PUNCT
ejde-761	237	4	now	now	ADV
ejde-761	237	5	,	,	PUNCT
ejde-761	237	6	assume	assume	VERB
ejde-761	237	7	that	that	SCONJ
ejde-761	237	8	for	for	ADP
ejde-761	237	9	any	any	DET
ejde-761	237	10	stationary	stationary	ADJ
ejde-761	237	11	perturbation	perturbation	NOUN
ejde-761	237	12	it	it	PRON
ejde-761	237	13	holds	hold	VERB
ejde-761	237	14	that	that	SCONJ
ejde-761	237	15	0	0	NUM
ejde-761	237	16	<	<	X
ejde-761	237	17	∥φ∥θ	∥φ∥θ	NOUN
ejde-761	237	18	≤	≤	ADJ
ejde-761	237	19	c.	c.	NOUN
ejde-761	237	20	in	in	ADP
ejde-761	237	21	addition	addition	NOUN
ejde-761	237	22	,	,	PUNCT
ejde-761	237	23	wt	wt	PROPN
ejde-761	237	24	=	=	SYM
ejde-761	237	25	f	f	PROPN
ejde-761	237	26	(	(	PUNCT
ejde-761	237	27	w	w	NOUN
ejde-761	237	28	)	)	PUNCT
ejde-761	237	29	,	,	PUNCT
ejde-761	237	30	(	(	PUNCT
ejde-761	237	31	4.15	4.15	NUM
ejde-761	237	32	)	)	PUNCT
ejde-761	237	33	where	where	SCONJ
ejde-761	237	34	f	f	PROPN
ejde-761	237	35	(	(	PUNCT
ejde-761	237	36	w	w	PROPN
ejde-761	237	37	)	)	PUNCT
ejde-761	237	38	=	=	SYM
ejde-761	237	39	−∆	−∆	PROPN
ejde-761	237	40	(	(	PUNCT
ejde-761	237	41	∑∞	∑∞	NOUN
ejde-761	237	42	k=0	k=0	PROPN
ejde-761	237	43	(	(	PUNCT
ejde-761	237	44	m	m	VERB
ejde-761	237	45	k	k	NOUN
ejde-761	237	46	)	)	PUNCT
ejde-761	237	47	(	(	PUNCT
ejde-761	237	48	∆w)m−k(∆φ)k∆(w	∆w)m−k(∆φ)k∆(w	NOUN
ejde-761	237	49	+	+	PROPN
ejde-761	237	50	φ	φ	NUM
ejde-761	237	51	)	)	PUNCT
ejde-761	237	52	)	)	PUNCT
ejde-761	238	1	+	+	CCONJ
ejde-761	238	2	∑∞	∑∞	X
ejde-761	238	3	j=0	j=0	PROPN
ejde-761	238	4	(	(	PUNCT
ejde-761	238	5	p	p	PROPN
ejde-761	238	6	j	j	PROPN
ejde-761	238	7	)	)	PUNCT
ejde-761	238	8	wp−jφj	wp−jφj	X
ejde-761	238	9	.	.	PUNCT
ejde-761	239	1	lemma	lemma	PROPN
ejde-761	239	2	4.3	4.3	NUM
ejde-761	239	3	.	.	PUNCT
ejde-761	240	1	the	the	DET
ejde-761	240	2	mapping	mapping	NOUN
ejde-761	240	3	f	f	X
ejde-761	240	4	:	:	PUNCT
ejde-761	240	5	hm	hm	INTJ
ejde-761	240	6	ρ	ρ	PROPN
ejde-761	240	7	→	→	SYM
ejde-761	240	8	l2	l2	NOUN
ejde-761	240	9	is	be	AUX
ejde-761	240	10	continuously	continuously	ADV
ejde-761	240	11	bounded	bound	VERB
ejde-761	240	12	.	.	PUNCT
ejde-761	241	1	in	in	ADP
ejde-761	241	2	addition	addition	NOUN
ejde-761	241	3	,	,	PUNCT
ejde-761	241	4	there	there	PRON
ejde-761	241	5	exist	exist	VERB
ejde-761	241	6	α0	α0	PROPN
ejde-761	241	7	>	>	X
ejde-761	241	8	0	0	NUM
ejde-761	241	9	,	,	PUNCT
ejde-761	241	10	k3	k3	VERB
ejde-761	241	11	>	>	X
ejde-761	241	12	0	0	PUNCT
ejde-761	241	13	and	and	CCONJ
ejde-761	241	14	α0	α0	ADJ
ejde-761	241	15	>	>	X
ejde-761	241	16	1	1	NUM
ejde-761	241	17	such	such	ADJ
ejde-761	241	18	that	that	SCONJ
ejde-761	242	1	∥f	∥f	PROPN
ejde-761	242	2	(	(	PUNCT
ejde-761	242	3	w)∥l2	w)∥l2	PROPN
ejde-761	242	4	≤	≤	NUM
ejde-761	242	5	k3∥w∥α0	k3∥w∥α0	PROPN
ejde-761	242	6	hm	hm	INTJ
ejde-761	243	1	ρ	ρ	PROPN
ejde-761	243	2	,	,	PUNCT
ejde-761	243	3	provided	provide	VERB
ejde-761	243	4	0	0	NUM
ejde-761	243	5	<	<	X
ejde-761	243	6	∥w∥hm	∥w∥hm	X
ejde-761	243	7	ρ	ρ	X
ejde-761	243	8	<	<	X
ejde-761	243	9	α0	α0	ADJ
ejde-761	243	10	.	.	PUNCT
ejde-761	244	1	proof	proof	NOUN
ejde-761	244	2	.	.	PUNCT
ejde-761	245	1	we	we	PRON
ejde-761	245	2	have	have	VERB
ejde-761	245	3	∥f	∥f	PROPN
ejde-761	245	4	(	(	PUNCT
ejde-761	245	5	w)∥l2	w)∥l2	PROPN
ejde-761	245	6	≤	≤	ADV
ejde-761	245	7	∥f	∥f	PROPN
ejde-761	245	8	(	(	PUNCT
ejde-761	245	9	w)∥θ	w)∥θ	NOUN
ejde-761	245	10	≤	≤	VERB
ejde-761	246	1	∞∑	∞∑	NUM
ejde-761	246	2	k=0	k=0	PROPN
ejde-761	246	3	(	(	PUNCT
ejde-761	246	4	m	m	VERB
ejde-761	246	5	k	k	NOUN
ejde-761	246	6	)	)	PUNCT
ejde-761	246	7	∥w∥m−k	∥w∥m−k	VERB
ejde-761	246	8	θ	θ	PROPN
ejde-761	246	9	∥φ∥kθ(∥w∥θ	∥φ∥kθ(∥w∥θ	PROPN
ejde-761	246	10	+	+	CCONJ
ejde-761	246	11	∥φ∥θ	∥φ∥θ	NOUN
ejde-761	246	12	)	)	PUNCT
ejde-761	247	1	+	+	CCONJ
ejde-761	247	2	∞∑	∞∑	NUM
ejde-761	247	3	j=0	j=0	PROPN
ejde-761	247	4	(	(	PUNCT
ejde-761	247	5	p	p	PROPN
ejde-761	247	6	j	j	PROPN
ejde-761	247	7	)	)	PUNCT
ejde-761	247	8	∥w∥p−j	∥w∥p−j	ADP
ejde-761	247	9	θ	θ	PROPN
ejde-761	247	10	∥φ∥jθ	∥φ∥jθ	ADJ
ejde-761	247	11	.	.	PUNCT
ejde-761	248	1	considering	consider	VERB
ejde-761	248	2	that	that	SCONJ
ejde-761	248	3	0	0	NUM
ejde-761	248	4	<	<	X
ejde-761	248	5	∥φ∥θ	∥φ∥θ	NOUN
ejde-761	248	6	≤	≤	NUM
ejde-761	248	7	c	c	NOUN
ejde-761	248	8	and	and	CCONJ
ejde-761	248	9	inequality	inequality	NOUN
ejde-761	248	10	(	(	PUNCT
ejde-761	248	11	3.8	3.8	NUM
ejde-761	248	12	)	)	PUNCT
ejde-761	248	13	,	,	PUNCT
ejde-761	248	14	∥f	∥f	PROPN
ejde-761	248	15	(	(	PUNCT
ejde-761	248	16	w)∥l2	w)∥l2	PROPN
ejde-761	248	17	≤	≤	ADV
ejde-761	248	18	∥f	∥f	PROPN
ejde-761	248	19	(	(	PUNCT
ejde-761	248	20	w)∥θ	w)∥θ	NOUN
ejde-761	248	21	≤	≤	VERB
ejde-761	248	22	∞∑	∞∑	NUM
ejde-761	248	23	k=0	k=0	PROPN
ejde-761	248	24	(	(	PUNCT
ejde-761	248	25	m	m	VERB
ejde-761	248	26	k	k	NOUN
ejde-761	248	27	)	)	PUNCT
ejde-761	248	28	2∥w∥m−k+1	2∥w∥m−k+1	NUM
ejde-761	248	29	hm	hm	INTJ
ejde-761	248	30	ρ	ρ	NOUN
ejde-761	248	31	ck+1	ck+1	ADV
ejde-761	248	32	+	+	CCONJ
ejde-761	248	33	∞∑	∞∑	NUM
ejde-761	248	34	j=0	j=0	PROPN
ejde-761	248	35	(	(	PUNCT
ejde-761	248	36	p	p	PROPN
ejde-761	248	37	j	j	PROPN
ejde-761	248	38	)	)	PUNCT
ejde-761	248	39	∥w∥p−j	∥w∥p−j	ADV
ejde-761	249	1	hm	hm	INTJ
ejde-761	249	2	ρ	ρ	NOUN
ejde-761	249	3	cj	cj	PROPN
ejde-761	249	4	.	.	PUNCT
ejde-761	250	1	(	(	PUNCT
ejde-761	250	2	4.16	4.16	NUM
ejde-761	250	3	)	)	PUNCT
ejde-761	250	4	12	12	NUM
ejde-761	250	5	j.	j.	PROPN
ejde-761	250	6	l.	l.	PROPN
ejde-761	250	7	díaz	díaz	PROPN
ejde-761	250	8	palencia	palencia	PROPN
ejde-761	250	9	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	250	10	for	for	ADP
ejde-761	250	11	m−	m−	PROPN
ejde-761	250	12	k	k	PROPN
ejde-761	251	1	+	+	CCONJ
ejde-761	251	2	1	1	X
ejde-761	251	3	>	>	X
ejde-761	251	4	p−	p−	PROPN
ejde-761	251	5	j	j	PROPN
ejde-761	251	6	,	,	PUNCT
ejde-761	251	7	the	the	DET
ejde-761	251	8	following	follow	VERB
ejde-761	251	9	holds	hold	VERB
ejde-761	251	10	∥f	∥f	PROPN
ejde-761	251	11	(	(	PUNCT
ejde-761	251	12	w)∥l2	w)∥l2	PROPN
ejde-761	251	13	≤	≤	ADV
ejde-761	251	14	∥f	∥f	PROPN
ejde-761	251	15	(	(	PUNCT
ejde-761	251	16	w)∥θ	w)∥θ	NOUN
ejde-761	251	17	≤	≤	VERB
ejde-761	252	1	∞∑	∞∑	NUM
ejde-761	252	2	k=0	k=0	PROPN
ejde-761	252	3	(	(	PUNCT
ejde-761	252	4	m	m	VERB
ejde-761	252	5	k	k	NOUN
ejde-761	252	6	)	)	PUNCT
ejde-761	252	7	3∥w∥m−k+1	3∥w∥m−k+1	NUM
ejde-761	252	8	hm	hm	INTJ
ejde-761	252	9	ρ	ρ	NUM
ejde-761	252	10	ck+1	ck+1	NOUN
ejde-761	252	11	=	=	PUNCT
ejde-761	252	12	k3∥w∥α0	k3∥w∥α0	NOUN
ejde-761	253	1	hm	hm	INTJ
ejde-761	253	2	ρ	ρ	PROPN
ejde-761	253	3	(	(	PUNCT
ejde-761	253	4	4.17	4.17	NUM
ejde-761	253	5	)	)	PUNCT
ejde-761	253	6	then	then	ADV
ejde-761	253	7	,	,	PUNCT
ejde-761	253	8	it	it	PRON
ejde-761	253	9	suffices	suffice	VERB
ejde-761	253	10	to	to	PART
ejde-761	253	11	consider	consider	VERB
ejde-761	253	12	k3	k3	ADJ
ejde-761	253	13	=	=	NOUN
ejde-761	253	14	∑∞	∑∞	NOUN
ejde-761	253	15	k=0	k=0	PROPN
ejde-761	253	16	(	(	PUNCT
ejde-761	253	17	m	m	VERB
ejde-761	253	18	k	k	NOUN
ejde-761	253	19	)	)	PUNCT
ejde-761	253	20	3ck+1	3ck+1	NUM
ejde-761	253	21	and	and	CCONJ
ejde-761	253	22	α0	α0	ADJ
ejde-761	253	23	=	=	SYM
ejde-761	253	24	max{m−	max{m−	PROPN
ejde-761	253	25	k+1	k+1	NOUN
ejde-761	253	26	}	}	PUNCT
ejde-761	253	27	>	>	X
ejde-761	253	28	1	1	X
ejde-761	253	29	.	.	PUNCT
ejde-761	253	30	considering	consider	VERB
ejde-761	253	31	that	that	SCONJ
ejde-761	253	32	p−	p−	PROPN
ejde-761	253	33	j	j	PROPN
ejde-761	253	34	>	>	X
ejde-761	253	35	m−	m−	PROPN
ejde-761	254	1	k	k	PROPN
ejde-761	255	1	+	+	CCONJ
ejde-761	255	2	1	1	NUM
ejde-761	255	3	,	,	PUNCT
ejde-761	255	4	we	we	PRON
ejde-761	255	5	have	have	VERB
ejde-761	255	6	∥f	∥f	PROPN
ejde-761	255	7	(	(	PUNCT
ejde-761	255	8	w)∥l2	w)∥l2	PROPN
ejde-761	255	9	≤	≤	ADV
ejde-761	255	10	∥f	∥f	PROPN
ejde-761	255	11	(	(	PUNCT
ejde-761	255	12	w)∥θ	w)∥θ	NOUN
ejde-761	255	13	≤	≤	NUM
ejde-761	255	14	∞∑	∞∑	NUM
ejde-761	255	15	j=0	j=0	PROPN
ejde-761	255	16	(	(	PUNCT
ejde-761	255	17	p	p	PROPN
ejde-761	255	18	j	j	PROPN
ejde-761	255	19	)	)	PUNCT
ejde-761	256	1	3∥w∥p−j	3∥w∥p−j	ADV
ejde-761	256	2	hm	hm	INTJ
ejde-761	256	3	ρ	ρ	NOUN
ejde-761	256	4	cj+1	cj+1	PROPN
ejde-761	256	5	=	=	SYM
ejde-761	256	6	k3∥w∥α0	k3∥w∥α0	PROPN
ejde-761	256	7	hm	hm	INTJ
ejde-761	256	8	ρ	ρ	PROPN
ejde-761	256	9	.	.	PUNCT
ejde-761	257	1	(	(	PUNCT
ejde-761	257	2	4.18	4.18	NUM
ejde-761	257	3	)	)	PUNCT
ejde-761	257	4	in	in	ADP
ejde-761	257	5	this	this	DET
ejde-761	257	6	case	case	NOUN
ejde-761	257	7	,	,	PUNCT
ejde-761	257	8	it	it	PRON
ejde-761	257	9	suffices	suffice	VERB
ejde-761	257	10	to	to	PART
ejde-761	257	11	admit	admit	VERB
ejde-761	257	12	k3	k3	VERB
ejde-761	257	13	=	=	PUNCT
ejde-761	257	14	∑∞	∑∞	X
ejde-761	257	15	j=0	j=0	PROPN
ejde-761	257	16	(	(	PUNCT
ejde-761	257	17	p	p	NOUN
ejde-761	257	18	j	j	PROPN
ejde-761	257	19	)	)	PUNCT
ejde-761	257	20	3cj+1	3cj+1	NUM
ejde-761	257	21	and	and	CCONJ
ejde-761	257	22	α0	α0	PROPN
ejde-761	257	23	=	=	SYM
ejde-761	257	24	max{p−	max{p−	PROPN
ejde-761	257	25	j	j	PROPN
ejde-761	257	26	}	}	PUNCT
ejde-761	257	27	>	>	X
ejde-761	258	1	1	1	X
ejde-761	258	2	.	.	PUNCT
ejde-761	259	1	the	the	DET
ejde-761	259	2	continuity	continuity	NOUN
ejde-761	259	3	can	can	AUX
ejde-761	259	4	be	be	AUX
ejde-761	259	5	shown	show	VERB
ejde-761	259	6	as	as	ADP
ejde-761	259	7	a	a	DET
ejde-761	259	8	consequence	consequence	NOUN
ejde-761	259	9	of	of	ADP
ejde-761	259	10	the	the	DET
ejde-761	259	11	proved	prove	VERB
ejde-761	259	12	inequalities	inequality	NOUN
ejde-761	259	13	by	by	ADP
ejde-761	259	14	considering	consider	VERB
ejde-761	259	15	a	a	DET
ejde-761	259	16	pair	pair	NOUN
ejde-761	259	17	of	of	ADP
ejde-761	259	18	sequences	sequence	NOUN
ejde-761	259	19	sufficiently	sufficiently	ADV
ejde-761	259	20	close	close	ADJ
ejde-761	259	21	.	.	PUNCT
ejde-761	260	1	this	this	PRON
ejde-761	260	2	can	can	AUX
ejde-761	260	3	be	be	AUX
ejde-761	260	4	done	do	VERB
ejde-761	260	5	by	by	ADP
ejde-761	260	6	standard	standard	ADJ
ejde-761	260	7	assessments	assessment	NOUN
ejde-761	260	8	.	.	PUNCT
ejde-761	261	1	□	□	PUNCT
ejde-761	261	2	consider	consider	VERB
ejde-761	261	3	the	the	DET
ejde-761	261	4	problem	problem	NOUN
ejde-761	261	5	wt	wt	ADP
ejde-761	261	6	=	=	SYM
ejde-761	261	7	−∆	−∆	NOUN
ejde-761	261	8	(	(	PUNCT
ejde-761	261	9	∞∑	∞∑	NUM
ejde-761	261	10	k=0	k=0	PROPN
ejde-761	261	11	(	(	PUNCT
ejde-761	261	12	m	m	VERB
ejde-761	261	13	k	k	NOUN
ejde-761	261	14	)	)	PUNCT
ejde-761	261	15	(	(	PUNCT
ejde-761	261	16	∆w)m−k(∆φ)k∆(w	∆w)m−k(∆φ)k∆(w	NOUN
ejde-761	261	17	+	+	CCONJ
ejde-761	261	18	φ	φ	NUM
ejde-761	261	19	)	)	PUNCT
ejde-761	261	20	)	)	PUNCT
ejde-761	262	1	+	+	CCONJ
ejde-761	262	2	∞∑	∞∑	NUM
ejde-761	262	3	j=0	j=0	PROPN
ejde-761	262	4	(	(	PUNCT
ejde-761	262	5	p	p	PROPN
ejde-761	262	6	j	j	PROPN
ejde-761	262	7	)	)	PUNCT
ejde-761	262	8	wp−jφj	wp−jφj	VERB
ejde-761	262	9	=	=	SYM
ejde-761	262	10	lw	lw	NOUN
ejde-761	262	11	+	+	NOUN
ejde-761	262	12	g(w	g(w	X
ejde-761	262	13	)	)	PUNCT
ejde-761	262	14	,	,	PUNCT
ejde-761	262	15	(	(	PUNCT
ejde-761	262	16	4.19	4.19	NUM
ejde-761	262	17	)	)	PUNCT
ejde-761	263	1	such	such	ADJ
ejde-761	263	2	that	that	DET
ejde-761	263	3	lw	lw	NOUN
ejde-761	263	4	=	=	PUNCT
ejde-761	263	5	−∆	−∆	PROPN
ejde-761	263	6	(	(	PUNCT
ejde-761	263	7	∞∑	∞∑	NUM
ejde-761	263	8	k=0	k=0	PROPN
ejde-761	263	9	(	(	PUNCT
ejde-761	263	10	m	m	VERB
ejde-761	263	11	k	k	NOUN
ejde-761	263	12	)	)	PUNCT
ejde-761	263	13	(	(	PUNCT
ejde-761	263	14	∆w)m−k(∆φ)k∆(w	∆w)m−k(∆φ)k∆(w	NOUN
ejde-761	263	15	+	+	PROPN
ejde-761	263	16	φ	φ	NUM
ejde-761	263	17	)	)	PUNCT
ejde-761	263	18	)	)	PUNCT
ejde-761	263	19	and	and	CCONJ
ejde-761	263	20	g(w	g(w	PROPN
ejde-761	263	21	)	)	PUNCT
ejde-761	263	22	=	=	PUNCT
ejde-761	263	23	∑∞	∑∞	NOUN
ejde-761	263	24	j=0	j=0	PROPN
ejde-761	263	25	(	(	PUNCT
ejde-761	263	26	p	p	PROPN
ejde-761	263	27	j	j	PROPN
ejde-761	263	28	)	)	PUNCT
ejde-761	263	29	wp−jφj	wp−jφj	X
ejde-761	263	30	.	.	PUNCT
ejde-761	264	1	based	base	VERB
ejde-761	264	2	on	on	ADP
ejde-761	264	3	this	this	PRON
ejde-761	264	4	,	,	PUNCT
ejde-761	264	5	we	we	PRON
ejde-761	264	6	consider	consider	VERB
ejde-761	264	7	the	the	DET
ejde-761	264	8	abstract	abstract	ADJ
ejde-761	264	9	evolution	evolution	PROPN
ejde-761	264	10	w(x	w(x	PROPN
ejde-761	264	11	,	,	PUNCT
ejde-761	264	12	t	t	PROPN
ejde-761	264	13	)	)	PUNCT
ejde-761	264	14	=	=	SYM
ejde-761	264	15	et	et	X
ejde-761	264	16	l	l	NOUN
ejde-761	264	17	w0(x	w0(x	PROPN
ejde-761	264	18	)	)	PUNCT
ejde-761	264	19	.	.	PUNCT
ejde-761	265	1	lemma	lemma	PROPN
ejde-761	265	2	4.4	4.4	NUM
ejde-761	265	3	.	.	PUNCT
ejde-761	266	1	l	l	NOUN
ejde-761	266	2	is	be	AUX
ejde-761	266	3	the	the	DET
ejde-761	266	4	infinitesimal	infinitesimal	ADJ
ejde-761	266	5	representation	representation	NOUN
ejde-761	266	6	of	of	ADP
ejde-761	266	7	a	a	DET
ejde-761	266	8	strongly	strongly	ADV
ejde-761	266	9	continuous	continuous	ADJ
ejde-761	266	10	semigroup	semigroup	NOUN
ejde-761	266	11	given	give	VERB
ejde-761	266	12	by	by	ADP
ejde-761	266	13	etl	etl	NOUN
ejde-761	266	14	that	that	PRON
ejde-761	266	15	satisfies∫	satisfies∫	NOUN
ejde-761	266	16	1	1	NUM
ejde-761	266	17	0	0	NUM
ejde-761	266	18	∥etl∥l2→hm	∥etl∥l2→hm	X
ejde-761	266	19	ρ	ρ	PROPN
ejde-761	266	20	=	=	SYM
ejde-761	266	21	k5	k5	PROPN
ejde-761	266	22	<	<	X
ejde-761	266	23	∞	∞	PROPN
ejde-761	266	24	,	,	PUNCT
ejde-761	266	25	∫	∫	PROPN
ejde-761	266	26	1	1	NUM
ejde-761	266	27	0	0	NUM
ejde-761	266	28	∥etl∥l2→hθ	∥etl∥l2→hθ	VERB
ejde-761	266	29	=	=	NOUN
ejde-761	266	30	k6	k6	NOUN
ejde-761	266	31	<	<	X
ejde-761	266	32	∞	∞	PROPN
ejde-761	266	33	(	(	PUNCT
ejde-761	266	34	4.20	4.20	NUM
ejde-761	266	35	)	)	PUNCT
ejde-761	266	36	proof	proof	NOUN
ejde-761	266	37	.	.	PUNCT
ejde-761	267	1	the	the	DET
ejde-761	267	2	proof	proof	NOUN
ejde-761	267	3	of	of	ADP
ejde-761	267	4	this	this	DET
ejde-761	267	5	lemma	lemma	PROPN
ejde-761	267	6	is	be	AUX
ejde-761	267	7	based	base	VERB
ejde-761	267	8	on	on	ADP
ejde-761	267	9	lemma	lemma	PROPN
ejde-761	267	10	3.1	3.1	NUM
ejde-761	267	11	.	.	PUNCT
ejde-761	268	1	then	then	ADV
ejde-761	268	2	∥w∥hm	∥w∥hm	X
ejde-761	268	3	ρ	ρ	PROPN
ejde-761	268	4	≤	≤	PROPN
ejde-761	268	5	∥w0∥hm	∥w0∥hm	PUNCT
ejde-761	268	6	ρ	ρ	NOUN
ejde-761	268	7	≤	≤	NUM
ejde-761	268	8	a0∥w0∥l2	a0∥w0∥l2	ADV
ejde-761	268	9	.	.	PUNCT
ejde-761	269	1	(	(	PUNCT
ejde-761	269	2	4.21	4.21	NUM
ejde-761	269	3	)	)	PUNCT
ejde-761	269	4	based	base	VERB
ejde-761	269	5	on	on	ADP
ejde-761	269	6	the	the	DET
ejde-761	269	7	abstract	abstract	ADJ
ejde-761	269	8	evolution	evolution	PROPN
ejde-761	269	9	w(x	w(x	PROPN
ejde-761	269	10	,	,	PUNCT
ejde-761	269	11	t	t	PROPN
ejde-761	269	12	)	)	PUNCT
ejde-761	269	13	=	=	PUNCT
ejde-761	270	1	etlw0(x	etlw0(x	PROPN
ejde-761	270	2	)	)	PUNCT
ejde-761	270	3	,	,	PUNCT
ejde-761	270	4	we	we	PRON
ejde-761	270	5	have	have	VERB
ejde-761	270	6	∥w∥hm	∥w∥hm	NOUN
ejde-761	270	7	ρ	ρ	PROPN
ejde-761	270	8	≤	≤	NUM
ejde-761	270	9	∥etl∥l2→hm	∥etl∥l2→hm	X
ejde-761	270	10	ρ	ρ	PROPN
ejde-761	270	11	∥w0∥l2	∥w0∥l2	PROPN
ejde-761	270	12	.	.	PUNCT
ejde-761	271	1	(	(	PUNCT
ejde-761	271	2	4.22	4.22	NUM
ejde-761	271	3	)	)	PUNCT
ejde-761	271	4	then	then	ADV
ejde-761	271	5	∫	∫	PROPN
ejde-761	271	6	1	1	NUM
ejde-761	271	7	0	0	NUM
ejde-761	271	8	∥etl∥l2→hm	∥etl∥l2→hm	X
ejde-761	271	9	ρ	ρ	PROPN
ejde-761	271	10	=	=	SYM
ejde-761	271	11	∫	∫	PROPN
ejde-761	271	12	1	1	NUM
ejde-761	271	13	0	0	NUM
ejde-761	271	14	a0	a0	PROPN
ejde-761	271	15	=	=	PROPN
ejde-761	271	16	k5	k5	PROPN
ejde-761	271	17	<	<	X
ejde-761	271	18	∞.	∞.	PROPN
ejde-761	271	19	(	(	PUNCT
ejde-761	271	20	4.23	4.23	NUM
ejde-761	271	21	)	)	PUNCT
ejde-761	271	22	it	it	PRON
ejde-761	271	23	is	be	AUX
ejde-761	271	24	easy	easy	ADJ
ejde-761	271	25	to	to	PART
ejde-761	271	26	check	check	VERB
ejde-761	271	27	that	that	SCONJ
ejde-761	271	28	the	the	DET
ejde-761	271	29	value	value	NOUN
ejde-761	271	30	of	of	ADP
ejde-761	271	31	a0	a0	PROPN
ejde-761	271	32	provides	provide	VERB
ejde-761	271	33	a	a	DET
ejde-761	271	34	finite	finite	ADJ
ejde-761	271	35	value	value	NOUN
ejde-761	271	36	of	of	ADP
ejde-761	271	37	k5	k5	PROPN
ejde-761	271	38	upon	upon	SCONJ
ejde-761	271	39	integration	integration	NOUN
ejde-761	271	40	in	in	ADP
ejde-761	271	41	t	t	PROPN
ejde-761	271	42	∈	∈	PROPN
ejde-761	271	43	(	(	PUNCT
ejde-761	271	44	0	0	NUM
ejde-761	271	45	,	,	PUNCT
ejde-761	271	46	1	1	NUM
ejde-761	271	47	]	]	PUNCT
ejde-761	271	48	.	.	PUNCT
ejde-761	272	1	operating	operate	VERB
ejde-761	272	2	similarly	similarly	ADV
ejde-761	272	3	,	,	PUNCT
ejde-761	272	4	it	it	PRON
ejde-761	272	5	is	be	AUX
ejde-761	272	6	possible	possible	ADJ
ejde-761	272	7	to	to	PART
ejde-761	272	8	conclude	conclude	VERB
ejde-761	272	9	on	on	ADP
ejde-761	272	10	the	the	DET
ejde-761	272	11	bound	bind	VERB
ejde-761	272	12	of	of	ADP
ejde-761	272	13	the	the	DET
ejde-761	272	14	abstract	abstract	ADJ
ejde-761	272	15	evolution	evolution	NOUN
ejde-761	272	16	in	in	ADP
ejde-761	272	17	hθ	hθ	PROPN
ejde-761	272	18	.	.	PUNCT
ejde-761	273	1	to	to	ADP
ejde-761	273	2	this	this	DET
ejde-761	273	3	end	end	NOUN
ejde-761	273	4	,	,	PUNCT
ejde-761	273	5	it	it	PRON
ejde-761	273	6	suffices	suffice	VERB
ejde-761	273	7	to	to	PART
ejde-761	273	8	consider	consider	VERB
ejde-761	273	9	the	the	DET
ejde-761	273	10	lemma	lemma	PROPN
ejde-761	273	11	3.1	3.1	NUM
ejde-761	273	12	along	along	ADP
ejde-761	273	13	with	with	ADP
ejde-761	273	14	the	the	DET
ejde-761	273	15	inequality	inequality	NOUN
ejde-761	273	16	(	(	PUNCT
ejde-761	273	17	3.8	3.8	NUM
ejde-761	273	18	)	)	PUNCT
ejde-761	273	19	,	,	PUNCT
ejde-761	273	20	∥w∥θ	∥w∥θ	NOUN
ejde-761	273	21	≤	≤	NOUN
ejde-761	273	22	σ	σ	NUM
ejde-761	273	23	∥w∥hm	∥w∥hm	PROPN
ejde-761	273	24	ρ	ρ	PROPN
ejde-761	273	25	≤	≤	PUNCT
ejde-761	273	26	σ∥w0∥hm	σ∥w0∥hm	NUM
ejde-761	273	27	ρ	ρ	PROPN
ejde-761	273	28	≤	≤	PROPN
ejde-761	273	29	σa0∥w0∥l2	σa0∥w0∥l2	NOUN
ejde-761	273	30	.	.	PUNCT
ejde-761	274	1	(	(	PUNCT
ejde-761	274	2	4.24	4.24	NUM
ejde-761	274	3	)	)	PUNCT
ejde-761	274	4	again	again	ADV
ejde-761	274	5	,	,	PUNCT
ejde-761	274	6	based	base	VERB
ejde-761	274	7	on	on	ADP
ejde-761	274	8	the	the	DET
ejde-761	274	9	abstract	abstract	ADJ
ejde-761	274	10	evolution	evolution	NOUN
ejde-761	274	11	,	,	PUNCT
ejde-761	274	12	∥w∥hθ	∥w∥hθ	NOUN
ejde-761	274	13	≤	≤	PUNCT
ejde-761	274	14	∥etl∥l2→hθ	∥etl∥l2→hθ	VERB
ejde-761	274	15	∥w0∥l2	∥w0∥l2	PROPN
ejde-761	274	16	,	,	PUNCT
ejde-761	274	17	(	(	PUNCT
ejde-761	274	18	4.25	4.25	NUM
ejde-761	274	19	)	)	PUNCT
ejde-761	274	20	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	274	21	instability	instability	NOUN
ejde-761	274	22	of	of	ADP
ejde-761	274	23	energy	energy	NOUN
ejde-761	274	24	solutions	solution	NOUN
ejde-761	274	25	13	13	NUM
ejde-761	274	26	so	so	SCONJ
ejde-761	274	27	that	that	SCONJ
ejde-761	274	28	∫	∫	PROPN
ejde-761	274	29	1	1	NUM
ejde-761	274	30	0	0	NUM
ejde-761	274	31	∥etl∥l2→hθ	∥etl∥l2→hθ	VERB
ejde-761	274	32	=	=	SYM
ejde-761	274	33	∫	∫	PROPN
ejde-761	274	34	1	1	NUM
ejde-761	274	35	0	0	NUM
ejde-761	274	36	a0σ	a0σ	ADJ
ejde-761	274	37	=	=	SYM
ejde-761	274	38	k6	k6	NOUN
ejde-761	274	39	<	<	X
ejde-761	274	40	∞	∞	PROPN
ejde-761	274	41	,	,	PUNCT
ejde-761	274	42	(	(	PUNCT
ejde-761	274	43	4.26	4.26	NUM
ejde-761	274	44	)	)	PUNCT
ejde-761	274	45	which	which	PRON
ejde-761	274	46	is	be	AUX
ejde-761	274	47	finite	finite	ADJ
ejde-761	274	48	upon	upon	SCONJ
ejde-761	274	49	integration	integration	NOUN
ejde-761	274	50	in	in	ADP
ejde-761	274	51	t	t	PROPN
ejde-761	274	52	∈	∈	PROPN
ejde-761	274	53	(	(	PUNCT
ejde-761	274	54	0	0	NUM
ejde-761	274	55	,	,	PUNCT
ejde-761	274	56	1	1	NUM
ejde-761	274	57	]	]	PUNCT
ejde-761	274	58	.	.	PUNCT
ejde-761	275	1	□	□	PUNCT
ejde-761	275	2	the	the	DET
ejde-761	275	3	next	next	ADJ
ejde-761	275	4	step	step	NOUN
ejde-761	275	5	is	be	AUX
ejde-761	275	6	to	to	PART
ejde-761	275	7	analyze	analyze	VERB
ejde-761	275	8	the	the	DET
ejde-761	275	9	spectrum	spectrum	NOUN
ejde-761	275	10	of	of	ADP
ejde-761	275	11	the	the	DET
ejde-761	275	12	operator	operator	NOUN
ejde-761	275	13	l	l	NOUN
ejde-761	275	14	(	(	PUNCT
ejde-761	275	15	as	as	SCONJ
ejde-761	275	16	defined	define	VERB
ejde-761	275	17	in	in	ADP
ejde-761	275	18	(	(	PUNCT
ejde-761	275	19	4.19	4.19	NUM
ejde-761	275	20	)	)	PUNCT
ejde-761	275	21	)	)	PUNCT
ejde-761	275	22	using	use	VERB
ejde-761	275	23	the	the	DET
ejde-761	275	24	norm	norm	NOUN
ejde-761	275	25	hθ	hθ	PROPN
ejde-761	275	26	.	.	PUNCT
ejde-761	276	1	lemma	lemma	PROPN
ejde-761	276	2	4.5	4.5	NUM
ejde-761	276	3	.	.	PUNCT
ejde-761	277	1	given	give	VERB
ejde-761	277	2	the	the	DET
ejde-761	277	3	condition	condition	NOUN
ejde-761	277	4	(	(	PUNCT
ejde-761	277	5	4.13	4.13	NUM
ejde-761	277	6	)	)	PUNCT
ejde-761	277	7	to	to	ADP
ejde-761	277	8	the	the	DET
ejde-761	277	9	perturbation	perturbation	NOUN
ejde-761	277	10	terms	term	NOUN
ejde-761	277	11	and	and	CCONJ
ejde-761	277	12	that	that	SCONJ
ejde-761	277	13	0	0	PUNCT
ejde-761	277	14	<	<	X
ejde-761	277	15	∥φ∥θ	∥φ∥θ	NOUN
ejde-761	277	16	≤	≤	NOUN
ejde-761	277	17	c	c	X
ejde-761	277	18	,	,	PUNCT
ejde-761	277	19	the	the	DET
ejde-761	277	20	spectrum	spectrum	NOUN
ejde-761	277	21	of	of	ADP
ejde-761	277	22	l	l	NOUN
ejde-761	277	23	(	(	PUNCT
ejde-761	277	24	see	see	VERB
ejde-761	277	25	(	(	PUNCT
ejde-761	277	26	4.19	4.19	NUM
ejde-761	277	27	)	)	PUNCT
ejde-761	277	28	)	)	PUNCT
ejde-761	277	29	in	in	ADP
ejde-761	277	30	hθ	hθ	PROPN
ejde-761	277	31	,	,	PUNCT
ejde-761	277	32	close	close	ADJ
ejde-761	277	33	to	to	ADP
ejde-761	277	34	the	the	DET
ejde-761	277	35	null	null	ADJ
ejde-761	277	36	solution	solution	NOUN
ejde-761	277	37	,	,	PUNCT
ejde-761	277	38	has	have	VERB
ejde-761	277	39	at	at	ADV
ejde-761	277	40	least	least	ADJ
ejde-761	277	41	an	an	DET
ejde-761	277	42	eigenvalue	eigenvalue	PROPN
ejde-761	277	43	(	(	PUNCT
ejde-761	277	44	ϕ	ϕ	NOUN
ejde-761	277	45	)	)	PUNCT
ejde-761	277	46	such	such	ADJ
ejde-761	277	47	that	that	PRON
ejde-761	277	48	re(ϕ	re(ϕ	NOUN
ejde-761	277	49	)	)	PUNCT
ejde-761	277	50	>	>	X
ejde-761	277	51	0	0	X
ejde-761	277	52	.	.	PUNCT
ejde-761	278	1	proof	proof	NOUN
ejde-761	278	2	.	.	PUNCT
ejde-761	279	1	this	this	DET
ejde-761	279	2	proposed	propose	VERB
ejde-761	279	3	lemma	lemma	PROPN
ejde-761	279	4	can	can	AUX
ejde-761	279	5	be	be	AUX
ejde-761	279	6	shown	show	VERB
ejde-761	279	7	through	through	ADP
ejde-761	279	8	the	the	DET
ejde-761	279	9	theory	theory	NOUN
ejde-761	279	10	of	of	ADP
ejde-761	279	11	evans	evans	PROPN
ejde-761	279	12	functions	function	NOUN
ejde-761	279	13	.	.	PUNCT
ejde-761	280	1	indeed	indeed	ADV
ejde-761	280	2	,	,	PUNCT
ejde-761	280	3	this	this	DET
ejde-761	280	4	theory	theory	NOUN
ejde-761	280	5	permits	permit	VERB
ejde-761	280	6	determining	determine	VERB
ejde-761	280	7	the	the	DET
ejde-761	280	8	location	location	NOUN
ejde-761	280	9	of	of	ADP
ejde-761	280	10	positive	positive	ADJ
ejde-761	280	11	eigenvalues	eigenvalue	NOUN
ejde-761	280	12	in	in	ADP
ejde-761	280	13	the	the	DET
ejde-761	280	14	proximity	proximity	NOUN
ejde-761	280	15	of	of	ADP
ejde-761	280	16	the	the	DET
ejde-761	280	17	null	null	ADJ
ejde-761	280	18	solution	solution	NOUN
ejde-761	280	19	.	.	PUNCT
ejde-761	281	1	the	the	DET
ejde-761	281	2	roots	root	NOUN
ejde-761	281	3	of	of	ADP
ejde-761	281	4	the	the	DET
ejde-761	281	5	evans	evans	PROPN
ejde-761	281	6	functions	function	NOUN
ejde-761	281	7	coincide	coincide	VERB
ejde-761	281	8	with	with	ADP
ejde-761	281	9	the	the	DET
ejde-761	281	10	characteristic	characteristic	ADJ
ejde-761	281	11	polynomial	polynomial	ADJ
ejde-761	281	12	roots	root	NOUN
ejde-761	281	13	of	of	ADP
ejde-761	281	14	a	a	DET
ejde-761	281	15	linearized	linearize	VERB
ejde-761	281	16	operator	operator	NOUN
ejde-761	281	17	close	close	ADV
ejde-761	281	18	to	to	ADP
ejde-761	281	19	the	the	DET
ejde-761	281	20	null	null	ADJ
ejde-761	281	21	solution	solution	NOUN
ejde-761	281	22	(	(	PUNCT
ejde-761	281	23	see	see	VERB
ejde-761	281	24	[	[	X
ejde-761	281	25	3	3	X
ejde-761	281	26	]	]	PUNCT
ejde-761	281	27	and	and	CCONJ
ejde-761	281	28	references	reference	NOUN
ejde-761	281	29	therein	therein	ADV
ejde-761	281	30	)	)	PUNCT
ejde-761	281	31	.	.	PUNCT
ejde-761	282	1	in	in	ADP
ejde-761	282	2	the	the	DET
ejde-761	282	3	presented	present	VERB
ejde-761	282	4	analysis	analysis	NOUN
ejde-761	282	5	,	,	PUNCT
ejde-761	282	6	the	the	DET
ejde-761	282	7	eigenvalues	eigenvalue	NOUN
ejde-761	282	8	are	be	AUX
ejde-761	282	9	obtained	obtain	VERB
ejde-761	282	10	by	by	ADP
ejde-761	282	11	using	use	VERB
ejde-761	282	12	the	the	DET
ejde-761	282	13	characteristic	characteristic	ADJ
ejde-761	282	14	polynomial	polynomial	NOUN
ejde-761	282	15	near	near	ADP
ejde-761	282	16	the	the	DET
ejde-761	282	17	equilibrium	equilibrium	NOUN
ejde-761	282	18	γ	γ	X
ejde-761	282	19	=	=	SYM
ejde-761	282	20	0	0	NUM
ejde-761	282	21	.	.	PUNCT
ejde-761	283	1	additionally	additionally	ADV
ejde-761	283	2	,	,	PUNCT
ejde-761	283	3	the	the	DET
ejde-761	283	4	particular	particular	ADJ
ejde-761	283	5	dynamics	dynamic	NOUN
ejde-761	283	6	,	,	PUNCT
ejde-761	283	7	affected	affect	VERB
ejde-761	283	8	by	by	ADP
ejde-761	283	9	the	the	DET
ejde-761	283	10	tw	tw	PROPN
ejde-761	283	11	speed	speed	PROPN
ejde-761	283	12	λ	λ	PROPN
ejde-761	283	13	,	,	PUNCT
ejde-761	283	14	are	be	AUX
ejde-761	283	15	assessed	assess	VERB
ejde-761	283	16	with	with	ADP
ejde-761	283	17	a	a	DET
ejde-761	283	18	homotopy	homotopy	NOUN
ejde-761	283	19	representation	representation	NOUN
ejde-761	283	20	.	.	PUNCT
ejde-761	284	1	to	to	ADP
ejde-761	284	2	this	this	DET
ejde-761	284	3	end	end	NOUN
ejde-761	284	4	,	,	PUNCT
ejde-761	284	5	a	a	DET
ejde-761	284	6	computational	computational	ADJ
ejde-761	284	7	exercise	exercise	NOUN
ejde-761	284	8	is	be	AUX
ejde-761	284	9	introduced	introduce	VERB
ejde-761	284	10	.	.	PUNCT
ejde-761	285	1	a	a	DET
ejde-761	285	2	first	first	ADJ
ejde-761	285	3	integral	integral	ADJ
ejde-761	285	4	can	can	AUX
ejde-761	285	5	be	be	AUX
ejde-761	285	6	derived	derive	VERB
ejde-761	285	7	in	in	ADP
ejde-761	285	8	the	the	DET
ejde-761	285	9	tw	tw	NOUN
ejde-761	285	10	problem	problem	NOUN
ejde-761	285	11	(	(	PUNCT
ejde-761	285	12	4.1	4.1	NUM
ejde-761	285	13	)	)	PUNCT
ejde-761	285	14	close	close	ADJ
ejde-761	285	15	to	to	ADP
ejde-761	285	16	the	the	DET
ejde-761	285	17	null	null	ADJ
ejde-761	285	18	solution	solution	NOUN
ejde-761	285	19	,	,	PUNCT
ejde-761	285	20	so	so	SCONJ
ejde-761	285	21	that	that	SCONJ
ejde-761	285	22	−λγ′	−λγ′	PROPN
ejde-761	285	23	=	=	SYM
ejde-761	286	1	−(|γ′′|mγ′′)′′	−(|γ′′|mγ′′)′′	PROPN
ejde-761	286	2	→	→	PUNCT
ejde-761	286	3	−λγ	−λγ	NOUN
ejde-761	286	4	=	=	PUNCT
ejde-761	286	5	−(|γ′′|mγ′′)′	−(|γ′′|mγ′′)′	PROPN
ejde-761	286	6	+	+	NUM
ejde-761	286	7	c1	c1	PROPN
ejde-761	286	8	,	,	PUNCT
ejde-761	286	9	(	(	PUNCT
ejde-761	286	10	4.27	4.27	NUM
ejde-761	286	11	)	)	PUNCT
ejde-761	286	12	where	where	SCONJ
ejde-761	286	13	the	the	DET
ejde-761	286	14	constant	constant	ADJ
ejde-761	286	15	c1	c1	NOUN
ejde-761	286	16	=	=	PROPN
ejde-761	286	17	0	0	NUM
ejde-761	286	18	for	for	ADP
ejde-761	286	19	the	the	DET
ejde-761	286	20	sake	sake	NOUN
ejde-761	286	21	of	of	ADP
ejde-761	286	22	simplicity	simplicity	NOUN
ejde-761	286	23	and	and	CCONJ
ejde-761	286	24	without	without	ADP
ejde-761	286	25	impacting	impact	VERB
ejde-761	286	26	the	the	DET
ejde-761	286	27	lemma	lemma	PROPN
ejde-761	286	28	results	result	NOUN
ejde-761	286	29	.	.	PUNCT
ejde-761	287	1	now	now	ADV
ejde-761	287	2	,	,	PUNCT
ejde-761	287	3	the	the	DET
ejde-761	287	4	problem	problem	NOUN
ejde-761	287	5	is	be	AUX
ejde-761	287	6	converted	convert	VERB
ejde-761	287	7	into	into	ADP
ejde-761	287	8	the	the	DET
ejde-761	287	9	matrix	matrix	NOUN
ejde-761	287	10	formulationγ0	formulationγ0	PROPN
ejde-761	287	11	γ2	γ2	PROPN
ejde-761	287	12	γ3	γ3	NOUN
ejde-761	287	13	′	′	PROPN
ejde-761	288	1	=	=	PUNCT
ejde-761	288	2			PROPN
ejde-761	288	3	0	0	NUM
ejde-761	288	4	1	1	NUM
ejde-761	288	5	0	0	NUM
ejde-761	288	6	0	0	NUM
ejde-761	288	7	0	0	NUM
ejde-761	288	8	1	1	NUM
ejde-761	288	9	λ	λ	NOUN
ejde-761	288	10	mγm−1	mγm−1	NOUN
ejde-761	288	11	3	3	NUM
ejde-761	288	12	+	+	NOUN
ejde-761	288	13	γm	γm	ADJ
ejde-761	288	14	3	3	NUM
ejde-761	288	15	0	0	NUM
ejde-761	288	16	0	0	NUM
ejde-761	288	17	γ0	γ0	NOUN
ejde-761	288	18	γ2	γ2	PROPN
ejde-761	288	19	γ3	γ3	NOUN
ejde-761	288	20			PROPN
ejde-761	288	21	(	(	PUNCT
ejde-761	288	22	4.28	4.28	NUM
ejde-761	288	23	)	)	PUNCT
ejde-761	288	24	note	note	NOUN
ejde-761	288	25	that	that	SCONJ
ejde-761	288	26	the	the	DET
ejde-761	288	27	characteristic	characteristic	ADJ
ejde-761	288	28	polynomial	polynomial	NOUN
ejde-761	288	29	(	(	PUNCT
ejde-761	288	30	in	in	ADP
ejde-761	288	31	the	the	DET
ejde-761	288	32	assumption	assumption	NOUN
ejde-761	288	33	that	that	SCONJ
ejde-761	288	34	γ3	γ3	NOUN
ejde-761	288	35	is	be	AUX
ejde-761	288	36	a	a	DET
ejde-761	288	37	free	free	ADJ
ejde-761	288	38	parameter	parameter	NOUN
ejde-761	288	39	)	)	PUNCT
ejde-761	288	40	for	for	ADP
ejde-761	288	41	the	the	DET
ejde-761	288	42	matrix	matrix	NOUN
ejde-761	288	43	is	be	AUX
ejde-761	288	44	q(ϕ	q(ϕ	PROPN
ejde-761	288	45	)	)	PUNCT
ejde-761	288	46	=	=	PUNCT
ejde-761	289	1	−ϕ3	−ϕ3	NOUN
ejde-761	290	1	+	+	PUNCT
ejde-761	290	2	λ	λ	X
ejde-761	290	3	mγm−1	mγm−1	NOUN
ejde-761	290	4	3	3	NUM
ejde-761	290	5	+	+	CCONJ
ejde-761	290	6	γm	γm	PRON
ejde-761	290	7	3	3	NUM
ejde-761	290	8	=	=	SYM
ejde-761	290	9	0	0	NUM
ejde-761	290	10	.	.	PUNCT
ejde-761	291	1	(	(	PUNCT
ejde-761	291	2	4.29	4.29	NUM
ejde-761	291	3	)	)	PUNCT
ejde-761	291	4	note	note	VERB
ejde-761	292	1	that	that	SCONJ
ejde-761	292	2	γ3	γ3	NOUN
ejde-761	292	3	=	=	SYM
ejde-761	292	4	γ′′	γ′′	PROPN
ejde-761	292	5	,	,	PUNCT
ejde-761	292	6	as	as	SCONJ
ejde-761	292	7	per	per	ADP
ejde-761	292	8	the	the	DET
ejde-761	292	9	standard	standard	ADJ
ejde-761	292	10	change	change	NOUN
ejde-761	292	11	of	of	ADP
ejde-761	292	12	variables	variable	NOUN
ejde-761	292	13	to	to	PART
ejde-761	292	14	build	build	VERB
ejde-761	292	15	the	the	DET
ejde-761	292	16	matrix	matrix	NOUN
ejde-761	292	17	representation	representation	NOUN
ejde-761	292	18	.	.	PUNCT
ejde-761	293	1	for	for	ADP
ejde-761	293	2	our	our	PRON
ejde-761	293	3	purposes	purpose	NOUN
ejde-761	293	4	(	(	PUNCT
ejde-761	293	5	i.e.	i.e.	X
ejde-761	293	6	to	to	PART
ejde-761	293	7	show	show	VERB
ejde-761	293	8	that	that	SCONJ
ejde-761	293	9	there	there	PRON
ejde-761	293	10	exists	exist	VERB
ejde-761	293	11	at	at	ADP
ejde-761	293	12	least	least	ADV
ejde-761	293	13	one	one	NUM
ejde-761	293	14	eigenvalue	eigenvalue	NOUN
ejde-761	293	15	with	with	ADP
ejde-761	293	16	positive	positive	ADJ
ejde-761	293	17	real	real	ADJ
ejde-761	293	18	part	part	NOUN
ejde-761	293	19	)	)	PUNCT
ejde-761	293	20	we	we	PRON
ejde-761	293	21	consider	consider	VERB
ejde-761	293	22	that	that	SCONJ
ejde-761	293	23	in	in	ADP
ejde-761	293	24	the	the	DET
ejde-761	293	25	asymptotic	asymptotic	ADJ
ejde-761	293	26	approximation	approximation	NOUN
ejde-761	293	27	to	to	ADP
ejde-761	293	28	the	the	DET
ejde-761	293	29	null	null	ADJ
ejde-761	293	30	state	state	NOUN
ejde-761	293	31	|γ3|	|γ3|	NOUN
ejde-761	293	32	≪	≪	ADJ
ejde-761	293	33	1	1	NUM
ejde-761	293	34	then	then	ADV
ejde-761	293	35	the	the	DET
ejde-761	293	36	following	follow	VERB
ejde-761	293	37	variable	variable	NOUN
ejde-761	293	38	υ	υ	PROPN
ejde-761	293	39	≫	≫	NOUN
ejde-761	293	40	1	1	NUM
ejde-761	293	41	is	be	AUX
ejde-761	293	42	introduced	introduce	VERB
ejde-761	293	43	,	,	PUNCT
ejde-761	293	44	−ϕ3	−ϕ3	PROPN
ejde-761	294	1	+	+	ADJ
ejde-761	294	2	υλ	υλ	NOUN
ejde-761	294	3	=	=	SYM
ejde-761	294	4	0	0	NUM
ejde-761	294	5	.	.	PUNCT
ejde-761	295	1	(	(	PUNCT
ejde-761	295	2	4.30	4.30	NUM
ejde-761	295	3	)	)	PUNCT
ejde-761	295	4	by	by	ADP
ejde-761	295	5	a	a	DET
ejde-761	295	6	standard	standard	ADJ
ejde-761	295	7	resolution	resolution	NOUN
ejde-761	295	8	of	of	ADP
ejde-761	295	9	the	the	DET
ejde-761	295	10	last	last	ADJ
ejde-761	295	11	characteristic	characteristic	ADJ
ejde-761	295	12	polynomial	polynomial	NOUN
ejde-761	295	13	,	,	PUNCT
ejde-761	295	14	it	it	PRON
ejde-761	295	15	is	be	AUX
ejde-761	295	16	easy	easy	ADJ
ejde-761	295	17	to	to	PART
ejde-761	295	18	conclude	conclude	VERB
ejde-761	295	19	on	on	ADP
ejde-761	295	20	the	the	DET
ejde-761	295	21	existence	existence	NOUN
ejde-761	295	22	of	of	ADP
ejde-761	295	23	at	at	ADV
ejde-761	295	24	least	least	ADV
ejde-761	295	25	one	one	NUM
ejde-761	295	26	eigenvalue	eigenvalue	NOUN
ejde-761	295	27	with	with	ADP
ejde-761	295	28	positive	positive	ADJ
ejde-761	295	29	real	real	ADJ
ejde-761	295	30	part	part	NOUN
ejde-761	295	31	.	.	PUNCT
ejde-761	296	1	in	in	ADP
ejde-761	296	2	addition	addition	NOUN
ejde-761	296	3	,	,	PUNCT
ejde-761	296	4	it	it	PRON
ejde-761	296	5	is	be	AUX
ejde-761	296	6	necessary	necessary	ADJ
ejde-761	296	7	to	to	PART
ejde-761	296	8	determine	determine	VERB
ejde-761	296	9	the	the	DET
ejde-761	296	10	effect	effect	NOUN
ejde-761	296	11	of	of	ADP
ejde-761	296	12	the	the	DET
ejde-761	296	13	tw	tw	NOUN
ejde-761	296	14	speed	speed	NOUN
ejde-761	296	15	.	.	PUNCT
ejde-761	297	1	to	to	ADP
ejde-761	297	2	this	this	DET
ejde-761	297	3	end	end	NOUN
ejde-761	297	4	,	,	PUNCT
ejde-761	297	5	the	the	DET
ejde-761	297	6	different	different	ADJ
ejde-761	297	7	homotopy	homotopy	NOUN
ejde-761	297	8	graphs	graph	NOUN
ejde-761	297	9	containing	contain	VERB
ejde-761	297	10	the	the	DET
ejde-761	297	11	eigenvalues	eigenvalue	NOUN
ejde-761	297	12	are	be	AUX
ejde-761	297	13	given	give	VERB
ejde-761	297	14	for	for	ADP
ejde-761	297	15	different	different	ADJ
ejde-761	297	16	values	value	NOUN
ejde-761	297	17	in	in	ADP
ejde-761	297	18	the	the	DET
ejde-761	297	19	tw	tw	NOUN
ejde-761	297	20	speed	speed	NOUN
ejde-761	297	21	(	(	PUNCT
ejde-761	297	22	see	see	VERB
ejde-761	297	23	figures	figure	NOUN
ejde-761	297	24	1	1	NUM
ejde-761	297	25	,	,	PUNCT
ejde-761	297	26	2	2	NUM
ejde-761	297	27	for	for	ADP
ejde-761	297	28	positive	positive	ADJ
ejde-761	297	29	tw	tw	NOUN
ejde-761	297	30	speeds	speed	NOUN
ejde-761	297	31	and	and	CCONJ
ejde-761	297	32	figures	figure	NOUN
ejde-761	297	33	3	3	NUM
ejde-761	297	34	,	,	PUNCT
ejde-761	297	35	4	4	NUM
ejde-761	297	36	for	for	ADP
ejde-761	297	37	negative	negative	ADJ
ejde-761	297	38	tw	tw	NOUN
ejde-761	297	39	speeds	speed	NOUN
ejde-761	297	40	)	)	PUNCT
ejde-761	297	41	.	.	PUNCT
ejde-761	298	1	□	□	PUNCT
ejde-761	298	2	14	14	NUM
ejde-761	298	3	j.	j.	PROPN
ejde-761	298	4	l.	l.	PROPN
ejde-761	298	5	díaz	díaz	PROPN
ejde-761	298	6	palencia	palencia	PROPN
ejde-761	298	7	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	298	8	figure	figure	NOUN
ejde-761	298	9	1	1	NUM
ejde-761	298	10	.	.	PUNCT
ejde-761	298	11	eigenvalues	eigenvalue	VERB
ejde-761	298	12	representations	representation	NOUN
ejde-761	298	13	in	in	ADP
ejde-761	298	14	the	the	DET
ejde-761	298	15	complex	complex	ADJ
ejde-761	298	16	plane	plane	NOUN
ejde-761	298	17	for	for	ADP
ejde-761	298	18	q(ϕ	q(ϕ	PROPN
ejde-761	298	19	)	)	PUNCT
ejde-761	298	20	roots	root	NOUN
ejde-761	298	21	.	.	PUNCT
ejde-761	299	1	the	the	DET
ejde-761	299	2	value	value	NOUN
ejde-761	299	3	of	of	ADP
ejde-761	299	4	υ	υ	NOUN
ejde-761	299	5	has	have	AUX
ejde-761	299	6	been	be	AUX
ejde-761	299	7	taken	take	VERB
ejde-761	299	8	arbitrary	arbitrary	ADJ
ejde-761	299	9	big	big	ADJ
ejde-761	299	10	.	.	PUNCT
ejde-761	300	1	note	note	VERB
ejde-761	300	2	that	that	SCONJ
ejde-761	300	3	υ	υ	PROPN
ejde-761	300	4	affects	affect	VERB
ejde-761	300	5	only	only	ADV
ejde-761	300	6	on	on	ADP
ejde-761	300	7	the	the	DET
ejde-761	300	8	scale	scale	NOUN
ejde-761	300	9	while	while	SCONJ
ejde-761	300	10	keeping	keep	VERB
ejde-761	300	11	the	the	DET
ejde-761	300	12	structure	structure	NOUN
ejde-761	300	13	of	of	ADP
ejde-761	300	14	the	the	DET
ejde-761	300	15	eigenvalues	eigenvalue	NOUN
ejde-761	300	16	(	(	PUNCT
ejde-761	300	17	one	one	NUM
ejde-761	300	18	with	with	ADP
ejde-761	300	19	positive	positive	ADJ
ejde-761	300	20	real	real	ADJ
ejde-761	300	21	part	part	NOUN
ejde-761	300	22	)	)	PUNCT
ejde-761	300	23	.	.	PUNCT
ejde-761	301	1	λ	λ	X
ejde-761	301	2	=	=	SYM
ejde-761	301	3	1	1	NUM
ejde-761	301	4	(	(	PUNCT
ejde-761	301	5	left	leave	VERB
ejde-761	301	6	)	)	PUNCT
ejde-761	301	7	and	and	CCONJ
ejde-761	301	8	λ	λ	X
ejde-761	301	9	=	=	VERB
ejde-761	301	10	10	10	NUM
ejde-761	301	11	(	(	PUNCT
ejde-761	301	12	right	right	NOUN
ejde-761	301	13	)	)	PUNCT
ejde-761	301	14	.	.	PUNCT
ejde-761	302	1	figure	figure	NOUN
ejde-761	302	2	2	2	NUM
ejde-761	302	3	.	.	PUNCT
ejde-761	302	4	eigenvalues	eigenvalue	VERB
ejde-761	302	5	representations	representation	NOUN
ejde-761	302	6	in	in	ADP
ejde-761	302	7	the	the	DET
ejde-761	302	8	complex	complex	ADJ
ejde-761	302	9	plane	plane	NOUN
ejde-761	302	10	for	for	ADP
ejde-761	302	11	q(ϕ	q(ϕ	PROPN
ejde-761	302	12	)	)	PUNCT
ejde-761	302	13	roots	root	NOUN
ejde-761	302	14	.	.	PUNCT
ejde-761	303	1	the	the	DET
ejde-761	303	2	value	value	NOUN
ejde-761	303	3	of	of	ADP
ejde-761	303	4	υ	υ	NOUN
ejde-761	303	5	has	have	AUX
ejde-761	303	6	been	be	AUX
ejde-761	303	7	taken	take	VERB
ejde-761	303	8	arbitrary	arbitrary	ADJ
ejde-761	303	9	big	big	ADJ
ejde-761	303	10	.	.	PUNCT
ejde-761	304	1	note	note	VERB
ejde-761	304	2	that	that	SCONJ
ejde-761	304	3	υ	υ	PROPN
ejde-761	304	4	affects	affect	VERB
ejde-761	304	5	only	only	ADV
ejde-761	304	6	on	on	ADP
ejde-761	304	7	the	the	DET
ejde-761	304	8	scale	scale	NOUN
ejde-761	304	9	while	while	SCONJ
ejde-761	304	10	keeping	keep	VERB
ejde-761	304	11	the	the	DET
ejde-761	304	12	structure	structure	NOUN
ejde-761	304	13	of	of	ADP
ejde-761	304	14	the	the	DET
ejde-761	304	15	eigenvalues	eigenvalue	NOUN
ejde-761	304	16	(	(	PUNCT
ejde-761	304	17	one	one	NUM
ejde-761	304	18	with	with	ADP
ejde-761	304	19	positive	positive	ADJ
ejde-761	304	20	real	real	ADJ
ejde-761	304	21	part	part	NOUN
ejde-761	304	22	)	)	PUNCT
ejde-761	304	23	.	.	PUNCT
ejde-761	305	1	λ	λ	X
ejde-761	305	2	=	=	NOUN
ejde-761	305	3	100	100	NUM
ejde-761	305	4	(	(	PUNCT
ejde-761	305	5	left	left	ADJ
ejde-761	305	6	)	)	PUNCT
ejde-761	305	7	and	and	CCONJ
ejde-761	305	8	λ	λ	X
ejde-761	305	9	=	=	SYM
ejde-761	305	10	1000	1000	NUM
ejde-761	305	11	(	(	PUNCT
ejde-761	305	12	right	right	NOUN
ejde-761	305	13	)	)	PUNCT
ejde-761	305	14	.	.	PUNCT
ejde-761	306	1	through	through	ADP
ejde-761	306	2	the	the	DET
ejde-761	306	3	provided	provide	VERB
ejde-761	306	4	series	series	NOUN
ejde-761	306	5	of	of	ADP
ejde-761	306	6	lemmas	lemmas	PROPN
ejde-761	306	7	,	,	PUNCT
ejde-761	306	8	it	it	PRON
ejde-761	306	9	was	be	AUX
ejde-761	306	10	shown	show	VERB
ejde-761	306	11	that	that	SCONJ
ejde-761	306	12	any	any	DET
ejde-761	306	13	oscillating	oscillate	VERB
ejde-761	306	14	energy	energy	NOUN
ejde-761	306	15	solution	solution	NOUN
ejde-761	306	16	is	be	AUX
ejde-761	306	17	bounded	bound	VERB
ejde-761	306	18	by	by	ADP
ejde-761	306	19	sobolev	sobolev	NOUN
ejde-761	306	20	norms	norm	NOUN
ejde-761	306	21	,	,	PUNCT
ejde-761	306	22	which	which	PRON
ejde-761	306	23	prevents	prevent	VERB
ejde-761	306	24	the	the	DET
ejde-761	306	25	formation	formation	NOUN
ejde-761	306	26	of	of	ADP
ejde-761	306	27	blow	blow	NOUN
ejde-761	306	28	-	-	PUNCT
ejde-761	306	29	up	up	ADP
ejde-761	306	30	profiles	profile	NOUN
ejde-761	306	31	.	.	PUNCT
ejde-761	307	1	additionally	additionally	ADV
ejde-761	307	2	,	,	PUNCT
ejde-761	307	3	the	the	DET
ejde-761	307	4	study	study	NOUN
ejde-761	307	5	of	of	ADP
ejde-761	307	6	the	the	DET
ejde-761	307	7	spectrum	spectrum	NOUN
ejde-761	307	8	of	of	ADP
ejde-761	307	9	the	the	DET
ejde-761	307	10	operator	operator	NOUN
ejde-761	307	11	l	l	NOUN
ejde-761	307	12	in	in	ADP
ejde-761	307	13	the	the	DET
ejde-761	307	14	hθ	hθ	PROPN
ejde-761	307	15	norm	norm	NOUN
ejde-761	307	16	indicated	indicate	VERB
ejde-761	307	17	the	the	DET
ejde-761	307	18	presence	presence	NOUN
ejde-761	307	19	of	of	ADP
ejde-761	307	20	at	at	ADV
ejde-761	307	21	least	least	ADV
ejde-761	307	22	one	one	NUM
ejde-761	307	23	eigenvalue	eigenvalue	NOUN
ejde-761	307	24	with	with	ADP
ejde-761	307	25	a	a	DET
ejde-761	307	26	positive	positive	ADJ
ejde-761	307	27	real	real	ADJ
ejde-761	307	28	part	part	NOUN
ejde-761	307	29	,	,	PUNCT
ejde-761	307	30	suggesting	suggest	VERB
ejde-761	307	31	an	an	DET
ejde-761	307	32	instability	instability	NOUN
ejde-761	307	33	in	in	ADP
ejde-761	307	34	the	the	DET
ejde-761	307	35	system	system	NOUN
ejde-761	307	36	.	.	PUNCT
ejde-761	308	1	this	this	DET
ejde-761	308	2	instability	instability	NOUN
ejde-761	308	3	is	be	AUX
ejde-761	308	4	influenced	influence	VERB
ejde-761	308	5	by	by	ADP
ejde-761	308	6	the	the	DET
ejde-761	308	7	speed	speed	NOUN
ejde-761	308	8	of	of	ADP
ejde-761	308	9	the	the	DET
ejde-761	308	10	traveling	travel	VERB
ejde-761	308	11	wave	wave	NOUN
ejde-761	308	12	,	,	PUNCT
ejde-761	308	13	as	as	SCONJ
ejde-761	308	14	demonstrated	demonstrate	VERB
ejde-761	308	15	by	by	ADP
ejde-761	308	16	the	the	DET
ejde-761	308	17	analysis	analysis	NOUN
ejde-761	308	18	of	of	ADP
ejde-761	308	19	eigenvalues	eigenvalue	NOUN
ejde-761	308	20	for	for	ADP
ejde-761	308	21	different	different	ADJ
ejde-761	308	22	tw	tw	NOUN
ejde-761	308	23	speeds	speed	NOUN
ejde-761	308	24	.	.	PUNCT
ejde-761	309	1	4.2	4.2	NUM
ejde-761	309	2	.	.	PUNCT
ejde-761	310	1	exact	exact	ADJ
ejde-761	310	2	travelling	travel	VERB
ejde-761	310	3	wave	wave	NOUN
ejde-761	310	4	profiles	profile	NOUN
ejde-761	310	5	and	and	CCONJ
ejde-761	310	6	characteristic	characteristic	ADJ
ejde-761	310	7	propagation	propagation	NOUN
ejde-761	310	8	speed	speed	NOUN
ejde-761	310	9	.	.	PUNCT
ejde-761	311	1	in	in	ADP
ejde-761	311	2	this	this	DET
ejde-761	311	3	section	section	NOUN
ejde-761	311	4	,	,	PUNCT
ejde-761	311	5	we	we	PRON
ejde-761	311	6	provide	provide	VERB
ejde-761	311	7	the	the	DET
ejde-761	311	8	tws	tws	NOUN
ejde-761	311	9	profiles	profile	NOUN
ejde-761	311	10	to	to	PART
ejde-761	311	11	validate	validate	VERB
ejde-761	311	12	the	the	DET
ejde-761	311	13	results	result	NOUN
ejde-761	311	14	obtained	obtain	VERB
ejde-761	311	15	in	in	ADP
ejde-761	311	16	the	the	DET
ejde-761	311	17	previous	previous	ADJ
ejde-761	311	18	section	section	NOUN
ejde-761	311	19	4.1	4.1	NUM
ejde-761	311	20	.	.	PUNCT
ejde-761	312	1	to	to	ADP
ejde-761	312	2	this	this	DET
ejde-761	312	3	end	end	NOUN
ejde-761	312	4	,	,	PUNCT
ejde-761	312	5	a	a	DET
ejde-761	312	6	numerical	numerical	ADJ
ejde-761	312	7	approach	approach	NOUN
ejde-761	312	8	has	have	AUX
ejde-761	312	9	been	be	AUX
ejde-761	312	10	followed	follow	VERB
ejde-761	312	11	using	use	VERB
ejde-761	312	12	the	the	DET
ejde-761	312	13	solver	solver	ADJ
ejde-761	312	14	bvp4c	bvp4c	NOUN
ejde-761	312	15	in	in	ADP
ejde-761	312	16	matlab	matlab	PROPN
ejde-761	312	17	.	.	PUNCT
ejde-761	313	1	this	this	DET
ejde-761	313	2	function	function	NOUN
ejde-761	313	3	consists	consist	VERB
ejde-761	313	4	on	on	ADP
ejde-761	313	5	a	a	DET
ejde-761	313	6	runge	runge	NOUN
ejde-761	313	7	-	-	PUNCT
ejde-761	313	8	kutta	kutta	NOUN
ejde-761	313	9	implicit	implicit	ADJ
ejde-761	313	10	algorithm	algorithm	NOUN
ejde-761	313	11	supported	support	VERB
ejde-761	313	12	by	by	ADP
ejde-761	313	13	interpolant	interpolant	NOUN
ejde-761	313	14	extensions	extension	NOUN
ejde-761	313	15	[	[	X
ejde-761	313	16	21	21	NUM
ejde-761	313	17	]	]	PUNCT
ejde-761	313	18	.	.	PUNCT
ejde-761	314	1	to	to	PART
ejde-761	314	2	build	build	VERB
ejde-761	314	3	the	the	DET
ejde-761	314	4	solution	solution	NOUN
ejde-761	314	5	,	,	PUNCT
ejde-761	314	6	the	the	DET
ejde-761	314	7	bvp4c	bvp4c	PROPN
ejde-761	314	8	requires	require	VERB
ejde-761	314	9	to	to	PART
ejde-761	314	10	solve	solve	VERB
ejde-761	314	11	a	a	DET
ejde-761	314	12	collocation	collocation	NOUN
ejde-761	314	13	method	method	NOUN
ejde-761	314	14	for	for	ADP
ejde-761	314	15	which	which	PRON
ejde-761	314	16	the	the	DET
ejde-761	314	17	conditions	condition	NOUN
ejde-761	314	18	at	at	ADP
ejde-761	314	19	ξ	ξ	X
ejde-761	314	20	→	→	SYM
ejde-761	314	21	−∞	−∞	PROPN
ejde-761	314	22	and	and	CCONJ
ejde-761	314	23	ξ	ξ	PROPN
ejde-761	314	24	→	→	SYM
ejde-761	314	25	∞	∞	PROPN
ejde-761	314	26	shall	shall	AUX
ejde-761	314	27	be	be	AUX
ejde-761	314	28	specified	specify	VERB
ejde-761	314	29	.	.	PUNCT
ejde-761	315	1	in	in	ADP
ejde-761	315	2	the	the	DET
ejde-761	315	3	presented	present	VERB
ejde-761	315	4	analysis	analysis	NOUN
ejde-761	315	5	,	,	PUNCT
ejde-761	315	6	the	the	DET
ejde-761	315	7	condition	condition	NOUN
ejde-761	315	8	at	at	ADP
ejde-761	315	9	−∞	−∞	X
ejde-761	315	10	is	be	AUX
ejde-761	315	11	admitted	admit	VERB
ejde-761	315	12	to	to	PART
ejde-761	315	13	be	be	AUX
ejde-761	315	14	positive	positive	ADJ
ejde-761	315	15	(	(	PUNCT
ejde-761	315	16	to	to	ADP
ejde-761	315	17	this	this	DET
ejde-761	315	18	end	end	NOUN
ejde-761	315	19	it	it	PRON
ejde-761	315	20	suffices	suffice	VERB
ejde-761	315	21	to	to	PART
ejde-761	315	22	consider	consider	VERB
ejde-761	315	23	γ(−∞	γ(−∞	NOUN
ejde-761	315	24	)	)	PUNCT
ejde-761	316	1	=	=	SYM
ejde-761	316	2	1	1	X
ejde-761	316	3	)	)	PUNCT
ejde-761	316	4	while	while	SCONJ
ejde-761	316	5	the	the	DET
ejde-761	316	6	condition	condition	NOUN
ejde-761	316	7	at	at	ADP
ejde-761	316	8	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	316	9	instability	instability	NOUN
ejde-761	316	10	of	of	ADP
ejde-761	316	11	energy	energy	NOUN
ejde-761	316	12	solutions	solution	NOUN
ejde-761	316	13	15	15	NUM
ejde-761	316	14	figure	figure	NOUN
ejde-761	316	15	3	3	NUM
ejde-761	316	16	.	.	PUNCT
ejde-761	316	17	eigenvalues	eigenvalue	VERB
ejde-761	316	18	representations	representation	NOUN
ejde-761	316	19	in	in	ADP
ejde-761	316	20	the	the	DET
ejde-761	316	21	complex	complex	ADJ
ejde-761	316	22	plane	plane	NOUN
ejde-761	316	23	for	for	ADP
ejde-761	316	24	q(ϕ	q(ϕ	PROPN
ejde-761	316	25	)	)	PUNCT
ejde-761	316	26	roots	root	NOUN
ejde-761	316	27	.	.	PUNCT
ejde-761	317	1	the	the	DET
ejde-761	317	2	value	value	NOUN
ejde-761	317	3	of	of	ADP
ejde-761	317	4	υ	υ	NOUN
ejde-761	317	5	has	have	AUX
ejde-761	317	6	been	be	AUX
ejde-761	317	7	taken	take	VERB
ejde-761	317	8	arbitrary	arbitrary	ADJ
ejde-761	317	9	big	big	ADJ
ejde-761	317	10	.	.	PUNCT
ejde-761	318	1	note	note	VERB
ejde-761	318	2	that	that	SCONJ
ejde-761	318	3	υ	υ	PROPN
ejde-761	318	4	affects	affect	VERB
ejde-761	318	5	only	only	ADV
ejde-761	318	6	on	on	ADP
ejde-761	318	7	the	the	DET
ejde-761	318	8	scale	scale	NOUN
ejde-761	318	9	while	while	SCONJ
ejde-761	318	10	keeping	keep	VERB
ejde-761	318	11	the	the	DET
ejde-761	318	12	structure	structure	NOUN
ejde-761	318	13	of	of	ADP
ejde-761	318	14	the	the	DET
ejde-761	318	15	eigenvalues	eigenvalue	NOUN
ejde-761	318	16	(	(	PUNCT
ejde-761	318	17	one	one	NUM
ejde-761	318	18	with	with	ADP
ejde-761	318	19	positive	positive	ADJ
ejde-761	318	20	real	real	ADJ
ejde-761	318	21	part	part	NOUN
ejde-761	318	22	)	)	PUNCT
ejde-761	318	23	.	.	PUNCT
ejde-761	319	1	λ	λ	NOUN
ejde-761	319	2	=	=	VERB
ejde-761	319	3	−1	−1	NOUN
ejde-761	319	4	(	(	PUNCT
ejde-761	319	5	left	left	ADJ
ejde-761	319	6	)	)	PUNCT
ejde-761	319	7	and	and	CCONJ
ejde-761	319	8	λ	λ	X
ejde-761	319	9	=	=	SYM
ejde-761	319	10	−10	−10	PROPN
ejde-761	319	11	(	(	PUNCT
ejde-761	319	12	right	right	ADJ
ejde-761	319	13	)	)	PUNCT
ejde-761	319	14	.	.	PUNCT
ejde-761	320	1	figure	figure	VERB
ejde-761	320	2	4	4	NUM
ejde-761	320	3	.	.	PUNCT
ejde-761	320	4	eigenvalues	eigenvalue	VERB
ejde-761	320	5	representations	representation	NOUN
ejde-761	320	6	in	in	ADP
ejde-761	320	7	the	the	DET
ejde-761	320	8	complex	complex	ADJ
ejde-761	320	9	plane	plane	NOUN
ejde-761	320	10	for	for	ADP
ejde-761	320	11	q(ϕ	q(ϕ	PROPN
ejde-761	320	12	)	)	PUNCT
ejde-761	320	13	roots	root	NOUN
ejde-761	320	14	.	.	PUNCT
ejde-761	321	1	the	the	DET
ejde-761	321	2	value	value	NOUN
ejde-761	321	3	of	of	ADP
ejde-761	321	4	υ	υ	NOUN
ejde-761	321	5	has	have	AUX
ejde-761	321	6	been	be	AUX
ejde-761	321	7	taken	take	VERB
ejde-761	321	8	arbitrary	arbitrary	ADJ
ejde-761	321	9	big	big	ADJ
ejde-761	321	10	.	.	PUNCT
ejde-761	322	1	note	note	VERB
ejde-761	322	2	that	that	SCONJ
ejde-761	322	3	υ	υ	PROPN
ejde-761	322	4	affects	affect	VERB
ejde-761	322	5	only	only	ADV
ejde-761	322	6	on	on	ADP
ejde-761	322	7	the	the	DET
ejde-761	322	8	scale	scale	NOUN
ejde-761	322	9	while	while	SCONJ
ejde-761	322	10	keeping	keep	VERB
ejde-761	322	11	the	the	DET
ejde-761	322	12	structure	structure	NOUN
ejde-761	322	13	of	of	ADP
ejde-761	322	14	the	the	DET
ejde-761	322	15	eigenvalues	eigenvalue	NOUN
ejde-761	322	16	(	(	PUNCT
ejde-761	322	17	one	one	NUM
ejde-761	322	18	with	with	ADP
ejde-761	322	19	positive	positive	ADJ
ejde-761	322	20	real	real	ADJ
ejde-761	322	21	part	part	NOUN
ejde-761	322	22	)	)	PUNCT
ejde-761	322	23	.	.	PUNCT
ejde-761	323	1	λ	λ	NOUN
ejde-761	323	2	=	=	SYM
ejde-761	323	3	−100	−100	PROPN
ejde-761	323	4	(	(	PUNCT
ejde-761	323	5	left	left	ADJ
ejde-761	323	6	)	)	PUNCT
ejde-761	323	7	and	and	CCONJ
ejde-761	323	8	λ	λ	X
ejde-761	323	9	=	=	SYM
ejde-761	323	10	−1000	−1000	PROPN
ejde-761	323	11	(	(	PUNCT
ejde-761	323	12	right	right	ADJ
ejde-761	323	13	)	)	PUNCT
ejde-761	323	14	.	.	PUNCT
ejde-761	324	1	∞	∞	PROPN
ejde-761	324	2	is	be	AUX
ejde-761	324	3	kept	keep	VERB
ejde-761	324	4	free	free	ADJ
ejde-761	324	5	.	.	PUNCT
ejde-761	325	1	this	this	DET
ejde-761	325	2	last	last	ADJ
ejde-761	325	3	condition	condition	NOUN
ejde-761	325	4	is	be	AUX
ejde-761	325	5	particularly	particularly	ADV
ejde-761	325	6	relevant	relevant	ADJ
ejde-761	325	7	to	to	PART
ejde-761	325	8	characterize	characterize	VERB
ejde-761	325	9	the	the	DET
ejde-761	325	10	null	null	ADJ
ejde-761	325	11	solution	solution	NOUN
ejde-761	325	12	as	as	ADP
ejde-761	325	13	an	an	DET
ejde-761	325	14	attractor	attractor	NOUN
ejde-761	325	15	.	.	PUNCT
ejde-761	326	1	the	the	DET
ejde-761	326	2	numerical	numerical	PROPN
ejde-761	326	3	exploration	exploration	NOUN
ejde-761	326	4	has	have	AUX
ejde-761	326	5	been	be	AUX
ejde-761	326	6	done	do	VERB
ejde-761	326	7	over	over	ADP
ejde-761	326	8	a	a	DET
ejde-761	326	9	large	large	ADJ
ejde-761	326	10	interval	interval	NOUN
ejde-761	326	11	in	in	ADP
ejde-761	326	12	ξ	ξ	PROPN
ejde-761	326	13	∈	∈	PROPN
ejde-761	326	14	[	[	X
ejde-761	326	15	−100	−100	NOUN
ejde-761	326	16	,	,	PUNCT
ejde-761	326	17	1000	1000	NUM
ejde-761	326	18	]	]	PUNCT
ejde-761	326	19	so	so	SCONJ
ejde-761	326	20	that	that	SCONJ
ejde-761	326	21	the	the	DET
ejde-761	326	22	problem	problem	NOUN
ejde-761	326	23	is	be	AUX
ejde-761	326	24	not	not	PART
ejde-761	326	25	governed	govern	VERB
ejde-761	326	26	by	by	ADP
ejde-761	326	27	the	the	DET
ejde-761	326	28	collocation	collocation	NOUN
ejde-761	326	29	method	method	NOUN
ejde-761	326	30	values	value	NOUN
ejde-761	326	31	at	at	ADP
ejde-761	326	32	−∞	−∞	X
ejde-761	326	33	and	and	CCONJ
ejde-761	326	34	∞.	∞.	PROPN
ejde-761	326	35	in	in	ADP
ejde-761	326	36	addition	addition	NOUN
ejde-761	326	37	,	,	PUNCT
ejde-761	326	38	and	and	CCONJ
ejde-761	326	39	to	to	PART
ejde-761	326	40	make	make	VERB
ejde-761	326	41	the	the	DET
ejde-761	326	42	problem	problem	NOUN
ejde-761	326	43	tractable	tractable	ADJ
ejde-761	326	44	,	,	PUNCT
ejde-761	326	45	dedicated	dedicated	ADJ
ejde-761	326	46	values	value	NOUN
ejde-761	326	47	have	have	AUX
ejde-761	326	48	been	be	AUX
ejde-761	326	49	introduced	introduce	VERB
ejde-761	326	50	for	for	ADP
ejde-761	326	51	the	the	DET
ejde-761	326	52	parameters	parameter	NOUN
ejde-761	326	53	involved	involve	VERB
ejde-761	326	54	in	in	ADP
ejde-761	326	55	problem	problem	NOUN
ejde-761	326	56	(	(	PUNCT
ejde-761	326	57	1.1	1.1	NUM
ejde-761	326	58	)	)	PUNCT
ejde-761	326	59	without	without	ADP
ejde-761	326	60	loss	loss	NOUN
ejde-761	326	61	of	of	ADP
ejde-761	326	62	generality	generality	NOUN
ejde-761	326	63	.	.	PUNCT
ejde-761	327	1	particularly	particularly	ADV
ejde-761	327	2	,	,	PUNCT
ejde-761	327	3	it	it	PRON
ejde-761	327	4	has	have	AUX
ejde-761	327	5	been	be	AUX
ejde-761	327	6	considered	consider	VERB
ejde-761	327	7	m	m	NOUN
ejde-761	327	8	=	=	NOUN
ejde-761	327	9	3	3	NUM
ejde-761	327	10	and	and	CCONJ
ejde-761	327	11	p	p	NOUN
ejde-761	327	12	=	=	ADJ
ejde-761	327	13	2	2	X
ejde-761	327	14	.	.	PUNCT
ejde-761	328	1	it	it	PRON
ejde-761	328	2	should	should	AUX
ejde-761	328	3	be	be	AUX
ejde-761	328	4	noted	note	VERB
ejde-761	328	5	that	that	SCONJ
ejde-761	328	6	the	the	DET
ejde-761	328	7	value	value	NOUN
ejde-761	328	8	of	of	ADP
ejde-761	328	9	m	m	PROPN
ejde-761	328	10	has	have	AUX
ejde-761	328	11	been	be	AUX
ejde-761	328	12	chosen	choose	VERB
ejde-761	328	13	as	as	ADP
ejde-761	328	14	an	an	DET
ejde-761	328	15	odd	odd	ADJ
ejde-761	328	16	number	number	NOUN
ejde-761	328	17	to	to	PART
ejde-761	328	18	complement	complement	VERB
ejde-761	328	19	the	the	DET
ejde-761	328	20	hypothesis	hypothesis	NOUN
ejde-761	328	21	in	in	ADP
ejde-761	328	22	lemma	lemma	PROPN
ejde-761	328	23	3.1	3.1	NUM
ejde-761	328	24	,	,	PUNCT
ejde-761	328	25	where	where	SCONJ
ejde-761	328	26	m	m	NOUN
ejde-761	328	27	is	be	AUX
ejde-761	328	28	assumed	assume	VERB
ejde-761	328	29	to	to	PART
ejde-761	328	30	be	be	AUX
ejde-761	328	31	even	even	ADV
ejde-761	328	32	.	.	PUNCT
ejde-761	329	1	this	this	DET
ejde-761	329	2	choice	choice	NOUN
ejde-761	329	3	is	be	AUX
ejde-761	329	4	made	make	VERB
ejde-761	329	5	to	to	PART
ejde-761	329	6	provide	provide	VERB
ejde-761	329	7	additional	additional	ADJ
ejde-761	329	8	information	information	NOUN
ejde-761	329	9	and	and	CCONJ
ejde-761	329	10	to	to	PART
ejde-761	329	11	explore	explore	VERB
ejde-761	329	12	the	the	DET
ejde-761	329	13	numerical	numerical	ADJ
ejde-761	329	14	behavior	behavior	NOUN
ejde-761	329	15	of	of	ADP
ejde-761	329	16	solutions	solution	NOUN
ejde-761	329	17	with	with	ADP
ejde-761	329	18	an	an	DET
ejde-761	329	19	odd	odd	ADJ
ejde-761	329	20	value	value	NOUN
ejde-761	329	21	of	of	ADP
ejde-761	329	22	m.	m.	NOUN
ejde-761	329	23	then	then	ADV
ejde-761	329	24	,	,	PUNCT
ejde-761	329	25	for	for	ADP
ejde-761	329	26	these	these	DET
ejde-761	329	27	values	value	NOUN
ejde-761	329	28	,	,	PUNCT
ejde-761	329	29	solutions	solution	NOUN
ejde-761	329	30	are	be	AUX
ejde-761	329	31	represented	represent	VERB
ejde-761	329	32	for	for	ADP
ejde-761	329	33	a	a	DET
ejde-761	329	34	wide	wide	ADJ
ejde-761	329	35	interval	interval	NOUN
ejde-761	329	36	of	of	ADP
ejde-761	329	37	tw	tw	NOUN
ejde-761	329	38	-	-	NOUN
ejde-761	329	39	speeds	speed	NOUN
ejde-761	329	40	.	.	PUNCT
ejde-761	330	1	it	it	PRON
ejde-761	330	2	is	be	AUX
ejde-761	330	3	possible	possible	ADJ
ejde-761	330	4	to	to	PART
ejde-761	330	5	check	check	VERB
ejde-761	330	6	that	that	SCONJ
ejde-761	330	7	in	in	ADP
ejde-761	330	8	all	all	DET
ejde-761	330	9	cases	case	NOUN
ejde-761	330	10	the	the	DET
ejde-761	330	11	null	null	ADJ
ejde-761	330	12	solution	solution	NOUN
ejde-761	330	13	acts	act	VERB
ejde-761	330	14	as	as	ADP
ejde-761	330	15	an	an	DET
ejde-761	330	16	attractor	attractor	NOUN
ejde-761	330	17	(	(	PUNCT
ejde-761	330	18	remind	remind	VERB
ejde-761	330	19	that	that	SCONJ
ejde-761	330	20	the	the	DET
ejde-761	330	21	collocation	collocation	NOUN
ejde-761	330	22	required	require	VERB
ejde-761	330	23	by	by	ADP
ejde-761	330	24	the	the	DET
ejde-761	330	25	bvp4c	bvp4c	PROPN
ejde-761	330	26	solver	solver	NOUN
ejde-761	330	27	is	be	AUX
ejde-761	330	28	kept	keep	VERB
ejde-761	330	29	free	free	ADJ
ejde-761	330	30	16	16	NUM
ejde-761	330	31	j.	j.	PROPN
ejde-761	330	32	l.	l.	PROPN
ejde-761	330	33	díaz	díaz	PROPN
ejde-761	330	34	palencia	palencia	PROPN
ejde-761	330	35	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	330	36	at	at	ADP
ejde-761	330	37	∞	∞	NUM
ejde-761	330	38	where	where	SCONJ
ejde-761	330	39	the	the	DET
ejde-761	330	40	null	null	ADJ
ejde-761	330	41	state	state	NOUN
ejde-761	330	42	occurs	occur	VERB
ejde-761	330	43	)	)	PUNCT
ejde-761	330	44	.	.	PUNCT
ejde-761	331	1	in	in	ADP
ejde-761	331	2	addition	addition	NOUN
ejde-761	331	3	,	,	PUNCT
ejde-761	331	4	the	the	DET
ejde-761	331	5	overall	overall	ADJ
ejde-761	331	6	instabilities	instability	NOUN
ejde-761	331	7	magnitude	magnitude	NOUN
ejde-761	331	8	increases	increase	NOUN
ejde-761	331	9	for	for	ADP
ejde-761	331	10	decreasing	decrease	VERB
ejde-761	331	11	values	value	NOUN
ejde-761	331	12	in	in	ADP
ejde-761	331	13	the	the	DET
ejde-761	331	14	tw	tw	NOUN
ejde-761	331	15	-	-	NOUN
ejde-761	331	16	speed	speed	NOUN
ejde-761	331	17	.	.	PUNCT
ejde-761	332	1	as	as	ADP
ejde-761	332	2	a	a	DET
ejde-761	332	3	consequence	consequence	NOUN
ejde-761	332	4	of	of	ADP
ejde-761	332	5	the	the	DET
ejde-761	332	6	exposed	expose	VERB
ejde-761	332	7	analysis	analysis	NOUN
ejde-761	332	8	,	,	PUNCT
ejde-761	332	9	it	it	PRON
ejde-761	332	10	is	be	AUX
ejde-761	332	11	concluded	conclude	VERB
ejde-761	332	12	that	that	SCONJ
ejde-761	332	13	it	it	PRON
ejde-761	332	14	is	be	AUX
ejde-761	332	15	not	not	PART
ejde-761	332	16	possible	possible	ADJ
ejde-761	332	17	to	to	PART
ejde-761	332	18	find	find	VERB
ejde-761	332	19	a	a	DET
ejde-761	332	20	suitable	suitable	ADJ
ejde-761	332	21	tw	tw	NOUN
ejde-761	332	22	-	-	NOUN
ejde-761	332	23	speed	speed	NOUN
ejde-761	332	24	for	for	ADP
ejde-761	332	25	which	which	PRON
ejde-761	332	26	a	a	DET
ejde-761	332	27	positive	positive	ADJ
ejde-761	332	28	inner	inner	ADJ
ejde-761	332	29	region	region	NOUN
ejde-761	332	30	can	can	AUX
ejde-761	332	31	be	be	AUX
ejde-761	332	32	shown	show	VERB
ejde-761	332	33	(	(	PUNCT
ejde-761	332	34	see	see	VERB
ejde-761	332	35	[	[	X
ejde-761	332	36	25	25	NUM
ejde-761	332	37	]	]	PUNCT
ejde-761	332	38	for	for	ADP
ejde-761	332	39	a	a	DET
ejde-761	332	40	complete	complete	ADJ
ejde-761	332	41	discussion	discussion	NOUN
ejde-761	332	42	)	)	PUNCT
ejde-761	332	43	impeding	impede	VERB
ejde-761	332	44	the	the	DET
ejde-761	332	45	possibility	possibility	NOUN
ejde-761	332	46	of	of	ADP
ejde-761	332	47	formulating	formulate	VERB
ejde-761	332	48	a	a	DET
ejde-761	332	49	maximal	maximal	ADJ
ejde-761	332	50	kernel	kernel	NOUN
ejde-761	332	51	with	with	ADP
ejde-761	332	52	purely	purely	ADV
ejde-761	332	53	monotone	monotone	ADJ
ejde-761	332	54	behaviour	behaviour	NOUN
ejde-761	332	55	.	.	PUNCT
ejde-761	333	1	even	even	ADV
ejde-761	333	2	further	far	ADV
ejde-761	333	3	,	,	PUNCT
ejde-761	333	4	it	it	PRON
ejde-761	333	5	is	be	AUX
ejde-761	333	6	not	not	PART
ejde-761	333	7	possible	possible	ADJ
ejde-761	333	8	to	to	PART
ejde-761	333	9	conclude	conclude	VERB
ejde-761	333	10	on	on	ADP
ejde-761	333	11	a	a	DET
ejde-761	333	12	tw	tw	NOUN
ejde-761	333	13	-	-	NOUN
ejde-761	333	14	speed	speed	NOUN
ejde-761	333	15	for	for	ADP
ejde-761	333	16	which	which	PRON
ejde-761	333	17	the	the	DET
ejde-761	333	18	tw	tw	NOUN
ejde-761	333	19	profile	profile	NOUN
ejde-761	333	20	is	be	AUX
ejde-761	333	21	positive	positive	ADJ
ejde-761	333	22	in	in	ADP
ejde-761	333	23	the	the	DET
ejde-761	333	24	whole	whole	ADJ
ejde-761	333	25	space	space	NOUN
ejde-761	333	26	as	as	ADP
ejde-761	333	27	in	in	ADP
ejde-761	333	28	the	the	DET
ejde-761	333	29	classical	classical	ADJ
ejde-761	333	30	order	order	NOUN
ejde-761	333	31	two	two	NUM
ejde-761	333	32	kpp	kpp	ADJ
ejde-761	333	33	-	-	NOUN
ejde-761	333	34	problem	problem	NOUN
ejde-761	333	35	[	[	X
ejde-761	333	36	32	32	NUM
ejde-761	333	37	]	]	PUNCT
ejde-761	333	38	.	.	PUNCT
ejde-761	334	1	other	other	ADJ
ejde-761	334	2	values	value	NOUN
ejde-761	334	3	of	of	ADP
ejde-761	334	4	m	m	VERB
ejde-761	334	5	have	have	AUX
ejde-761	334	6	also	also	ADV
ejde-761	334	7	been	be	AUX
ejde-761	334	8	considered	consider	VERB
ejde-761	334	9	(	(	PUNCT
ejde-761	334	10	even	even	ADV
ejde-761	334	11	and	and	CCONJ
ejde-761	334	12	odd	odd	ADJ
ejde-761	334	13	)	)	PUNCT
ejde-761	334	14	in	in	ADP
ejde-761	334	15	the	the	DET
ejde-761	334	16	analysis	analysis	NOUN
ejde-761	334	17	,	,	PUNCT
ejde-761	334	18	but	but	CCONJ
ejde-761	334	19	the	the	DET
ejde-761	334	20	conclusions	conclusion	NOUN
ejde-761	334	21	remain	remain	VERB
ejde-761	334	22	the	the	DET
ejde-761	334	23	same	same	ADJ
ejde-761	334	24	.	.	PUNCT
ejde-761	335	1	regardless	regardless	ADV
ejde-761	335	2	of	of	ADP
ejde-761	335	3	whether	whether	SCONJ
ejde-761	335	4	m	m	NOUN
ejde-761	335	5	is	be	AUX
ejde-761	335	6	chosen	choose	VERB
ejde-761	335	7	to	to	PART
ejde-761	335	8	be	be	AUX
ejde-761	335	9	odd	odd	ADJ
ejde-761	335	10	or	or	CCONJ
ejde-761	335	11	even	even	ADV
ejde-761	335	12	,	,	PUNCT
ejde-761	335	13	the	the	DET
ejde-761	335	14	null	null	ADJ
ejde-761	335	15	solution	solution	NOUN
ejde-761	335	16	consistently	consistently	ADV
ejde-761	335	17	acts	act	VERB
ejde-761	335	18	as	as	ADP
ejde-761	335	19	an	an	DET
ejde-761	335	20	attractor	attractor	NOUN
ejde-761	335	21	,	,	PUNCT
ejde-761	335	22	and	and	CCONJ
ejde-761	335	23	the	the	DET
ejde-761	335	24	overall	overall	ADJ
ejde-761	335	25	magnitude	magnitude	NOUN
ejde-761	335	26	of	of	ADP
ejde-761	335	27	instabilities	instability	NOUN
ejde-761	335	28	increases	increase	VERB
ejde-761	335	29	as	as	ADP
ejde-761	335	30	the	the	DET
ejde-761	335	31	tw	tw	NOUN
ejde-761	335	32	-	-	PUNCT
ejde-761	335	33	speed	speed	NOUN
ejde-761	335	34	decreases	decrease	NOUN
ejde-761	335	35	.	.	PUNCT
ejde-761	336	1	hence	hence	ADV
ejde-761	336	2	,	,	PUNCT
ejde-761	336	3	this	this	DET
ejde-761	336	4	particular	particular	ADJ
ejde-761	336	5	behaviour	behaviour	NOUN
ejde-761	336	6	in	in	ADP
ejde-761	336	7	the	the	DET
ejde-761	336	8	solutions	solution	NOUN
ejde-761	336	9	is	be	AUX
ejde-761	336	10	not	not	PART
ejde-761	336	11	dependent	dependent	ADJ
ejde-761	336	12	on	on	ADP
ejde-761	336	13	the	the	DET
ejde-761	336	14	specific	specific	ADJ
ejde-761	336	15	choice	choice	NOUN
ejde-761	336	16	of	of	ADP
ejde-761	336	17	m.	m.	NOUN
ejde-761	336	18	note	note	NOUN
ejde-761	336	19	that	that	SCONJ
ejde-761	336	20	these	these	DET
ejde-761	336	21	additional	additional	ADJ
ejde-761	336	22	results	result	NOUN
ejde-761	336	23	are	be	AUX
ejde-761	336	24	not	not	PART
ejde-761	336	25	included	include	VERB
ejde-761	336	26	in	in	ADP
ejde-761	336	27	this	this	DET
ejde-761	336	28	work	work	NOUN
ejde-761	336	29	to	to	PART
ejde-761	336	30	maintain	maintain	VERB
ejde-761	336	31	conciseness	conciseness	NOUN
ejde-761	336	32	.	.	PUNCT
ejde-761	337	1	figure	figure	NOUN
ejde-761	337	2	5	5	NUM
ejde-761	337	3	.	.	PUNCT
ejde-761	337	4	solution	solution	NOUN
ejde-761	337	5	profiles	profile	NOUN
ejde-761	337	6	for	for	ADP
ejde-761	337	7	low	low	ADJ
ejde-761	337	8	values	value	NOUN
ejde-761	337	9	of	of	ADP
ejde-761	337	10	tw	tw	NOUN
ejde-761	337	11	-	-	NOUN
ejde-761	337	12	speed	speed	NOUN
ejde-761	337	13	.	.	PUNCT
ejde-761	338	1	note	note	VERB
ejde-761	338	2	that	that	SCONJ
ejde-761	338	3	for	for	SCONJ
ejde-761	338	4	tw	tw	NOUN
ejde-761	338	5	-	-	PUNCT
ejde-761	338	6	speed	speed	NOUN
ejde-761	338	7	values	value	NOUN
ejde-761	338	8	close	close	ADJ
ejde-761	338	9	to	to	ADP
ejde-761	338	10	zero	zero	NUM
ejde-761	338	11	,	,	PUNCT
ejde-761	338	12	the	the	DET
ejde-761	338	13	null	null	ADJ
ejde-761	338	14	solution	solution	NOUN
ejde-761	338	15	does	do	AUX
ejde-761	338	16	not	not	PART
ejde-761	338	17	behave	behave	VERB
ejde-761	338	18	as	as	ADP
ejde-761	338	19	an	an	DET
ejde-761	338	20	attractor	attractor	NOUN
ejde-761	338	21	.	.	PUNCT
ejde-761	339	1	the	the	DET
ejde-761	339	2	increasing	increasing	NOUN
ejde-761	339	3	of	of	ADP
ejde-761	339	4	the	the	DET
ejde-761	339	5	tw	tw	NOUN
ejde-761	339	6	-	-	NOUN
ejde-761	339	7	speed	speed	NOUN
ejde-761	339	8	(	(	PUNCT
ejde-761	339	9	up	up	ADP
ejde-761	339	10	to	to	ADP
ejde-761	339	11	a	a	DET
ejde-761	339	12	moderate	moderate	ADJ
ejde-761	339	13	value	value	NOUN
ejde-761	339	14	of	of	ADP
ejde-761	339	15	0.1	0.1	NUM
ejde-761	339	16	)	)	PUNCT
ejde-761	339	17	stabilizes	stabilize	VERB
ejde-761	339	18	the	the	DET
ejde-761	339	19	attractor	attractor	NOUN
ejde-761	339	20	behaviour	behaviour	NOUN
ejde-761	339	21	of	of	ADP
ejde-761	339	22	the	the	DET
ejde-761	339	23	null	null	ADJ
ejde-761	339	24	critical	critical	ADJ
ejde-761	339	25	point	point	NOUN
ejde-761	339	26	.	.	PUNCT
ejde-761	340	1	figure	figure	NOUN
ejde-761	340	2	6	6	NUM
ejde-761	340	3	.	.	PUNCT
ejde-761	341	1	for	for	ADP
ejde-761	341	2	increasing	increase	VERB
ejde-761	341	3	values	value	NOUN
ejde-761	341	4	in	in	ADP
ejde-761	341	5	the	the	DET
ejde-761	341	6	tw	tw	NOUN
ejde-761	341	7	-	-	NOUN
ejde-761	341	8	speed	speed	NOUN
ejde-761	341	9	,	,	PUNCT
ejde-761	341	10	it	it	PRON
ejde-761	341	11	is	be	AUX
ejde-761	341	12	possible	possible	ADJ
ejde-761	341	13	to	to	PART
ejde-761	341	14	conclude	conclude	VERB
ejde-761	341	15	on	on	ADP
ejde-761	341	16	the	the	DET
ejde-761	341	17	attractor	attractor	NOUN
ejde-761	341	18	properties	property	NOUN
ejde-761	341	19	of	of	ADP
ejde-761	341	20	the	the	DET
ejde-761	341	21	null	null	ADJ
ejde-761	341	22	solution	solution	NOUN
ejde-761	341	23	.	.	PUNCT
ejde-761	342	1	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	342	2	instability	instability	NOUN
ejde-761	342	3	of	of	ADP
ejde-761	342	4	energy	energy	NOUN
ejde-761	342	5	solutions	solution	NOUN
ejde-761	342	6	17	17	NUM
ejde-761	342	7	5	5	NUM
ejde-761	342	8	.	.	PUNCT
ejde-761	342	9	scaling	scale	VERB
ejde-761	342	10	invariance	invariance	NOUN
ejde-761	342	11	and	and	CCONJ
ejde-761	342	12	symmetry	symmetry	NOUN
ejde-761	342	13	we	we	PRON
ejde-761	342	14	continue	continue	VERB
ejde-761	342	15	our	our	PRON
ejde-761	342	16	computation	computation	NOUN
ejde-761	342	17	of	of	ADP
ejde-761	342	18	solutions	solution	NOUN
ejde-761	342	19	by	by	ADP
ejde-761	342	20	considering	consider	VERB
ejde-761	342	21	symmetries	symmetry	NOUN
ejde-761	342	22	in	in	ADP
ejde-761	342	23	the	the	DET
ejde-761	342	24	equation	equation	NOUN
ejde-761	342	25	(	(	PUNCT
ejde-761	342	26	1.1	1.1	NUM
ejde-761	342	27	)	)	PUNCT
ejde-761	342	28	.	.	PUNCT
ejde-761	343	1	lemma	lemma	PROPN
ejde-761	343	2	5.1	5.1	NUM
ejde-761	343	3	.	.	PUNCT
ejde-761	344	1	the	the	DET
ejde-761	344	2	equation	equation	NOUN
ejde-761	344	3	(	(	PUNCT
ejde-761	344	4	1.1	1.1	NUM
ejde-761	344	5	)	)	PUNCT
ejde-761	344	6	possesses	possess	VERB
ejde-761	344	7	self	self	NOUN
ejde-761	344	8	-	-	PUNCT
ejde-761	344	9	similar	similar	ADJ
ejde-761	344	10	solutions	solution	NOUN
ejde-761	344	11	under	under	ADP
ejde-761	344	12	the	the	DET
ejde-761	344	13	scaling	scaling	ADJ
ejde-761	344	14	transformation	transformation	NOUN
ejde-761	344	15	u(x	u(x	NOUN
ejde-761	344	16	,	,	PUNCT
ejde-761	344	17	t	t	PROPN
ejde-761	344	18	)	)	PUNCT
ejde-761	344	19	→	→	SYM
ejde-761	344	20	λu(λbx	λu(λbx	NOUN
ejde-761	344	21	,	,	PUNCT
ejde-761	344	22	λct	λct	PROPN
ejde-761	344	23	)	)	PUNCT
ejde-761	344	24	with	with	ADP
ejde-761	344	25	the	the	DET
ejde-761	344	26	exponents	exponent	NOUN
ejde-761	344	27	b	b	PROPN
ejde-761	344	28	=	=	SYM
ejde-761	344	29	m−	m−	PROPN
ejde-761	344	30	(	(	PUNCT
ejde-761	344	31	p−	p−	NOUN
ejde-761	344	32	1	1	NUM
ejde-761	344	33	)	)	PUNCT
ejde-761	344	34	2(m+	2(m+	NUM
ejde-761	344	35	1	1	NUM
ejde-761	344	36	)	)	PUNCT
ejde-761	344	37	,	,	PUNCT
ejde-761	345	1	c	c	NOUN
ejde-761	345	2	=	=	SYM
ejde-761	345	3	1−	1−	NUM
ejde-761	346	1	p.	p.	NOUN
ejde-761	346	2	under	under	ADP
ejde-761	346	3	this	this	DET
ejde-761	346	4	transformation	transformation	NOUN
ejde-761	346	5	,	,	PUNCT
ejde-761	346	6	the	the	DET
ejde-761	346	7	original	original	ADJ
ejde-761	346	8	equation	equation	NOUN
ejde-761	346	9	remains	remain	VERB
ejde-761	346	10	invariant	invariant	ADJ
ejde-761	346	11	,	,	PUNCT
ejde-761	346	12	ensuring	ensure	VERB
ejde-761	346	13	the	the	DET
ejde-761	346	14	existence	existence	NOUN
ejde-761	346	15	of	of	ADP
ejde-761	346	16	self	self	NOUN
ejde-761	346	17	-	-	PUNCT
ejde-761	346	18	similar	similar	ADJ
ejde-761	346	19	solutions	solution	NOUN
ejde-761	346	20	.	.	PUNCT
ejde-761	347	1	proof	proof	NOUN
ejde-761	347	2	.	.	PUNCT
ejde-761	348	1	we	we	PRON
ejde-761	348	2	start	start	VERB
ejde-761	348	3	by	by	ADP
ejde-761	348	4	considering	consider	VERB
ejde-761	348	5	the	the	DET
ejde-761	348	6	scaling	scale	VERB
ejde-761	348	7	transformation	transformation	NOUN
ejde-761	348	8	:	:	PUNCT
ejde-761	348	9	u(x	u(x	PROPN
ejde-761	348	10	,	,	PUNCT
ejde-761	348	11	t	t	PROPN
ejde-761	348	12	)	)	PUNCT
ejde-761	348	13	→	→	SYM
ejde-761	348	14	λu(λbx	λu(λbx	NOUN
ejde-761	348	15	,	,	PUNCT
ejde-761	348	16	λct	λct	PROPN
ejde-761	348	17	)	)	PUNCT
ejde-761	348	18	.	.	PUNCT
ejde-761	349	1	applying	apply	VERB
ejde-761	349	2	this	this	DET
ejde-761	349	3	transformation	transformation	NOUN
ejde-761	349	4	to	to	ADP
ejde-761	349	5	each	each	DET
ejde-761	349	6	term	term	NOUN
ejde-761	349	7	in	in	ADP
ejde-761	349	8	the	the	DET
ejde-761	349	9	equation	equation	NOUN
ejde-761	349	10	,	,	PUNCT
ejde-761	349	11	we	we	PRON
ejde-761	349	12	obtain	obtain	VERB
ejde-761	349	13	ut	ut	PROPN
ejde-761	349	14	→	→	SYM
ejde-761	349	15	λ	λ	PROPN
ejde-761	349	16	∂	∂	NOUN
ejde-761	349	17	∂t	∂t	PROPN
ejde-761	349	18	u(λbx	u(λbx	PROPN
ejde-761	349	19	,	,	PUNCT
ejde-761	349	20	λct	λct	PROPN
ejde-761	349	21	)	)	PUNCT
ejde-761	349	22	=	=	SYM
ejde-761	350	1	λλ−c	λλ−c	NUM
ejde-761	350	2	∂	∂	NUM
ejde-761	350	3	∂(λct	∂(λct	NOUN
ejde-761	350	4	)	)	PUNCT
ejde-761	350	5	u(λbx	u(λbx	PROPN
ejde-761	350	6	,	,	PUNCT
ejde-761	350	7	λct	λct	PROPN
ejde-761	350	8	)	)	PUNCT
ejde-761	350	9	=	=	PUNCT
ejde-761	351	1	λ1−cut	λ1−cut	VERB
ejde-761	351	2	,	,	PUNCT
ejde-761	351	3	∆u	∆u	PROPN
ejde-761	351	4	→	→	SYM
ejde-761	351	5	λ1−2b∆u	λ1−2b∆u	NOUN
ejde-761	351	6	,	,	PUNCT
ejde-761	351	7	−∆(|∆u|m∆u	−∆(|∆u|m∆u	ADJ
ejde-761	351	8	)	)	PUNCT
ejde-761	351	9	→	→	SYM
ejde-761	351	10	λ1+m(1−2b)−2b∆(|∆u|m∆u	λ1+m(1−2b)−2b∆(|∆u|m∆u	PROPN
ejde-761	351	11	)	)	PUNCT
ejde-761	351	12	,	,	PUNCT
ejde-761	351	13	|u|p−1u	|u|p−1u	INTJ
ejde-761	351	14	→	→	SYM
ejde-761	351	15	λp|u|p−1u	λp|u|p−1u	NOUN
ejde-761	351	16	.	.	PUNCT
ejde-761	352	1	to	to	PART
ejde-761	352	2	ensure	ensure	VERB
ejde-761	352	3	the	the	DET
ejde-761	352	4	equation	equation	NOUN
ejde-761	352	5	is	be	AUX
ejde-761	352	6	invariant	invariant	ADJ
ejde-761	352	7	under	under	ADP
ejde-761	352	8	the	the	DET
ejde-761	352	9	scaling	scale	VERB
ejde-761	352	10	transformation	transformation	NOUN
ejde-761	352	11	,	,	PUNCT
ejde-761	352	12	the	the	DET
ejde-761	352	13	exponents	exponent	NOUN
ejde-761	352	14	of	of	ADP
ejde-761	352	15	λ	λ	PROPN
ejde-761	352	16	must	must	AUX
ejde-761	352	17	be	be	AUX
ejde-761	352	18	equal	equal	ADJ
ejde-761	352	19	for	for	ADP
ejde-761	352	20	each	each	DET
ejde-761	352	21	term	term	NOUN
ejde-761	352	22	.	.	PUNCT
ejde-761	353	1	this	this	PRON
ejde-761	353	2	leads	lead	VERB
ejde-761	353	3	to	to	ADP
ejde-761	353	4	the	the	DET
ejde-761	353	5	system	system	NOUN
ejde-761	353	6	of	of	ADP
ejde-761	353	7	equations	equation	NOUN
ejde-761	353	8	1−	1−	NUM
ejde-761	353	9	c	c	NOUN
ejde-761	353	10	=	=	SYM
ejde-761	353	11	p	p	NOUN
ejde-761	353	12	,	,	PUNCT
ejde-761	353	13	1−	1−	NUM
ejde-761	353	14	c	c	NOUN
ejde-761	353	15	=	=	SYM
ejde-761	353	16	1	1	NUM
ejde-761	353	17	+	+	NOUN
ejde-761	353	18	m(1−	m(1−	ADJ
ejde-761	353	19	2b)−	2b)−	NUM
ejde-761	353	20	2b	2b	NOUN
ejde-761	353	21	.	.	PUNCT
ejde-761	354	1	from	from	ADP
ejde-761	354	2	the	the	DET
ejde-761	354	3	first	first	ADJ
ejde-761	354	4	equation	equation	NOUN
ejde-761	354	5	,	,	PUNCT
ejde-761	354	6	c	c	PROPN
ejde-761	354	7	=	=	SYM
ejde-761	354	8	1−	1−	NUM
ejde-761	354	9	p.	p.	NOUN
ejde-761	354	10	then	then	ADV
ejde-761	354	11	substituting	substitute	VERB
ejde-761	354	12	into	into	ADP
ejde-761	354	13	the	the	DET
ejde-761	354	14	second	second	ADJ
ejde-761	354	15	equation	equation	NOUN
ejde-761	354	16	,	,	PUNCT
ejde-761	354	17	we	we	PRON
ejde-761	354	18	have	have	VERB
ejde-761	354	19	1−	1−	NUM
ejde-761	354	20	(	(	PUNCT
ejde-761	354	21	1−	1−	NUM
ejde-761	354	22	p	p	NOUN
ejde-761	354	23	)	)	PUNCT
ejde-761	354	24	=	=	SYM
ejde-761	355	1	1	1	NUM
ejde-761	356	1	+	+	NOUN
ejde-761	356	2	m(1−	m(1−	ADJ
ejde-761	356	3	2b)−	2b)−	NUM
ejde-761	356	4	2b	2b	NOUN
ejde-761	356	5	,	,	PUNCT
ejde-761	356	6	and	and	CCONJ
ejde-761	356	7	p	p	NOUN
ejde-761	356	8	=	=	SYM
ejde-761	356	9	1	1	NUM
ejde-761	356	10	+	+	NOUN
ejde-761	356	11	m(1−	m(1−	ADJ
ejde-761	356	12	2b)−	2b)−	NUM
ejde-761	356	13	2b	2b	NOUN
ejde-761	356	14	,	,	PUNCT
ejde-761	356	15	simplifying	simplify	VERB
ejde-761	356	16	p−	p−	NOUN
ejde-761	356	17	1	1	NUM
ejde-761	356	18	=	=	SYM
ejde-761	356	19	m(1−	m(1−	NOUN
ejde-761	356	20	2b)−	2b)−	NUM
ejde-761	356	21	2b	2b	NOUN
ejde-761	356	22	,	,	PUNCT
ejde-761	356	23	p−	p−	NOUN
ejde-761	356	24	1	1	NUM
ejde-761	356	25	=	=	SYM
ejde-761	356	26	m−	m−	PROPN
ejde-761	356	27	b(2m+	b(2m+	NOUN
ejde-761	356	28	2	2	NUM
ejde-761	356	29	)	)	PUNCT
ejde-761	356	30	,	,	PUNCT
ejde-761	356	31	b(2m+	b(2m+	NOUN
ejde-761	356	32	2	2	NUM
ejde-761	356	33	)	)	PUNCT
ejde-761	357	1	=	=	SYM
ejde-761	357	2	m−	m−	PROPN
ejde-761	357	3	(	(	PUNCT
ejde-761	357	4	p−	p−	NOUN
ejde-761	357	5	1	1	NUM
ejde-761	357	6	)	)	PUNCT
ejde-761	357	7	,	,	PUNCT
ejde-761	357	8	b	b	X
ejde-761	357	9	=	=	SYM
ejde-761	357	10	m−	m−	PROPN
ejde-761	357	11	(	(	PUNCT
ejde-761	357	12	p−	p−	NOUN
ejde-761	357	13	1	1	NUM
ejde-761	357	14	)	)	PUNCT
ejde-761	357	15	2(m+	2(m+	NUM
ejde-761	357	16	1	1	NUM
ejde-761	357	17	)	)	PUNCT
ejde-761	357	18	.	.	PUNCT
ejde-761	358	1	thus	thus	ADV
ejde-761	358	2	,	,	PUNCT
ejde-761	358	3	we	we	PRON
ejde-761	358	4	have	have	VERB
ejde-761	358	5	b	b	NOUN
ejde-761	358	6	=	=	SYM
ejde-761	358	7	m−	m−	PROPN
ejde-761	358	8	(	(	PUNCT
ejde-761	358	9	p−	p−	NOUN
ejde-761	358	10	1	1	NUM
ejde-761	358	11	)	)	PUNCT
ejde-761	358	12	2(m+	2(m+	NUM
ejde-761	358	13	1	1	NUM
ejde-761	358	14	)	)	PUNCT
ejde-761	358	15	,	,	PUNCT
ejde-761	359	1	c	c	NOUN
ejde-761	359	2	=	=	SYM
ejde-761	359	3	1−	1−	NUM
ejde-761	359	4	p.	p.	NOUN
ejde-761	359	5	this	this	PRON
ejde-761	359	6	completes	complete	VERB
ejde-761	359	7	the	the	DET
ejde-761	359	8	proof	proof	NOUN
ejde-761	359	9	that	that	SCONJ
ejde-761	359	10	the	the	DET
ejde-761	359	11	scaling	scale	VERB
ejde-761	359	12	transformation	transformation	NOUN
ejde-761	359	13	leaves	leave	VERB
ejde-761	359	14	the	the	DET
ejde-761	359	15	original	original	ADJ
ejde-761	359	16	equation	equation	NOUN
ejde-761	359	17	invariant	invariant	ADJ
ejde-761	359	18	,	,	PUNCT
ejde-761	359	19	hence	hence	ADV
ejde-761	359	20	ensuring	ensure	VERB
ejde-761	359	21	the	the	DET
ejde-761	359	22	existence	existence	NOUN
ejde-761	359	23	of	of	ADP
ejde-761	359	24	self	self	NOUN
ejde-761	359	25	-	-	PUNCT
ejde-761	359	26	similar	similar	ADJ
ejde-761	359	27	solutions	solution	NOUN
ejde-761	359	28	.	.	PUNCT
ejde-761	360	1	□	□	PUNCT
ejde-761	360	2	18	18	NUM
ejde-761	360	3	j.	j.	PROPN
ejde-761	360	4	l.	l.	PROPN
ejde-761	360	5	díaz	díaz	PROPN
ejde-761	360	6	palencia	palencia	PROPN
ejde-761	360	7	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	360	8	next	next	ADV
ejde-761	360	9	,	,	PUNCT
ejde-761	360	10	we	we	PRON
ejde-761	360	11	construct	construct	VERB
ejde-761	360	12	the	the	DET
ejde-761	360	13	self	self	NOUN
ejde-761	360	14	-	-	PUNCT
ejde-761	360	15	similar	similar	ADJ
ejde-761	360	16	solutions	solution	NOUN
ejde-761	360	17	.	.	PUNCT
ejde-761	361	1	we	we	PRON
ejde-761	361	2	assume	assume	VERB
ejde-761	361	3	a	a	DET
ejde-761	361	4	self	self	NOUN
ejde-761	361	5	-	-	PUNCT
ejde-761	361	6	similar	similar	ADJ
ejde-761	361	7	form	form	NOUN
ejde-761	361	8	u(x	u(x	NOUN
ejde-761	361	9	,	,	PUNCT
ejde-761	361	10	t	t	NOUN
ejde-761	361	11	)	)	PUNCT
ejde-761	361	12	=	=	PUNCT
ejde-761	362	1	(	(	PUNCT
ejde-761	362	2	t	t	PROPN
ejde-761	362	3	−	−	PROPN
ejde-761	362	4	t)−αf	t)−αf	PROPN
ejde-761	362	5	(	(	PUNCT
ejde-761	362	6	x	x	X
ejde-761	362	7	(	(	PUNCT
ejde-761	362	8	t	t	NOUN
ejde-761	362	9	−	−	PROPN
ejde-761	362	10	t)β	t)β	NOUN
ejde-761	362	11	)	)	PUNCT
ejde-761	362	12	,	,	PUNCT
ejde-761	362	13	where	where	SCONJ
ejde-761	362	14	t	t	PROPN
ejde-761	362	15	is	be	AUX
ejde-761	362	16	the	the	DET
ejde-761	362	17	blow	blow	NOUN
ejde-761	362	18	-	-	PUNCT
ejde-761	362	19	up	up	ADP
ejde-761	362	20	time	time	NOUN
ejde-761	362	21	,	,	PUNCT
ejde-761	362	22	and	and	CCONJ
ejde-761	362	23	α	α	NOUN
ejde-761	362	24	and	and	CCONJ
ejde-761	362	25	β	β	PROPN
ejde-761	362	26	are	be	AUX
ejde-761	362	27	constants	constant	NOUN
ejde-761	362	28	to	to	PART
ejde-761	362	29	be	be	AUX
ejde-761	362	30	determined	determine	VERB
ejde-761	362	31	.	.	PUNCT
ejde-761	363	1	by	by	ADP
ejde-761	363	2	comparing	compare	VERB
ejde-761	363	3	this	this	DET
ejde-761	363	4	form	form	NOUN
ejde-761	363	5	with	with	ADP
ejde-761	363	6	the	the	DET
ejde-761	363	7	scaling	scale	VERB
ejde-761	363	8	transformation	transformation	NOUN
ejde-761	363	9	,	,	PUNCT
ejde-761	363	10	we	we	PRON
ejde-761	363	11	set	set	VERB
ejde-761	363	12	α	α	NOUN
ejde-761	363	13	=	=	SYM
ejde-761	363	14	1	1	NUM
ejde-761	363	15	c	c	NOUN
ejde-761	363	16	,	,	PUNCT
ejde-761	363	17	β	β	X
ejde-761	363	18	=	=	PUNCT
ejde-761	363	19	b	b	PROPN
ejde-761	363	20	c	c	NOUN
ejde-761	363	21	.	.	PUNCT
ejde-761	364	1	from	from	ADP
ejde-761	364	2	c	c	NOUN
ejde-761	364	3	=	=	SYM
ejde-761	364	4	1−	1−	NUM
ejde-761	364	5	p	p	NOUN
ejde-761	364	6	,	,	PUNCT
ejde-761	364	7	we	we	PRON
ejde-761	364	8	have	have	VERB
ejde-761	364	9	α	α	NOUN
ejde-761	364	10	=	=	SYM
ejde-761	364	11	1	1	NUM
ejde-761	364	12	1−	1−	NUM
ejde-761	364	13	p	p	NOUN
ejde-761	364	14	,	,	PUNCT
ejde-761	364	15	β	β	X
ejde-761	364	16	=	=	SYM
ejde-761	364	17	m−(p−1	m−(p−1	PROPN
ejde-761	364	18	)	)	PUNCT
ejde-761	364	19	2(m+1	2(m+1	NUM
ejde-761	364	20	)	)	PUNCT
ejde-761	364	21	1−	1−	NUM
ejde-761	365	1	p	p	NOUN
ejde-761	365	2	=	=	X
ejde-761	365	3	m−	m−	PROPN
ejde-761	365	4	(	(	PUNCT
ejde-761	365	5	p−	p−	NOUN
ejde-761	365	6	1	1	NUM
ejde-761	365	7	)	)	PUNCT
ejde-761	365	8	2(m+	2(m+	NUM
ejde-761	365	9	1)(1−	1)(1−	NUM
ejde-761	365	10	p	p	NOUN
ejde-761	365	11	)	)	PUNCT
ejde-761	365	12	.	.	PUNCT
ejde-761	366	1	we	we	PRON
ejde-761	366	2	now	now	ADV
ejde-761	366	3	compute	compute	VERB
ejde-761	366	4	the	the	DET
ejde-761	366	5	derivatives	derivative	NOUN
ejde-761	366	6	.	.	PUNCT
ejde-761	367	1	ut	ut	PROPN
ejde-761	367	2	=	=	PROPN
ejde-761	367	3	α(t	α(t	PROPN
ejde-761	367	4	−	−	ADP
ejde-761	367	5	t)−α−1f(ξ	t)−α−1f(ξ	NOUN
ejde-761	367	6	)	)	PUNCT
ejde-761	368	1	+	+	CCONJ
ejde-761	368	2	βξ(t	βξ(t	X
ejde-761	368	3	−	−	NOUN
ejde-761	368	4	t)−α−1f	t)−α−1f	SYM
ejde-761	368	5	′(ξ	′(ξ	NOUN
ejde-761	368	6	)	)	PUNCT
ejde-761	368	7	,	,	PUNCT
ejde-761	368	8	uxx	uxx	X
ejde-761	368	9	=	=	SYM
ejde-761	368	10	(	(	PUNCT
ejde-761	368	11	t	t	PROPN
ejde-761	368	12	−	−	NOUN
ejde-761	369	1	t)−α−2βf	t)−α−2βf	PUNCT
ejde-761	369	2	′′(ξ	′′(ξ	PROPN
ejde-761	369	3	)	)	PUNCT
ejde-761	369	4	.	.	PUNCT
ejde-761	370	1	for	for	ADP
ejde-761	370	2	the	the	DET
ejde-761	370	3	nonlinear	nonlinear	ADJ
ejde-761	370	4	term	term	NOUN
ejde-761	370	5	,	,	PUNCT
ejde-761	370	6	we	we	PRON
ejde-761	370	7	have	have	VERB
ejde-761	370	8	|uxx|muxx	|uxx|muxx	VERB
ejde-761	370	9	=	=	SYM
ejde-761	370	10	(	(	PUNCT
ejde-761	370	11	t	t	NOUN
ejde-761	370	12	−	−	PROPN
ejde-761	370	13	t)(−α−2β)(m+1)|f	t)(−α−2β)(m+1)|f	PROPN
ejde-761	370	14	′′(ξ)|mf	′′(ξ)|mf	NOUN
ejde-761	370	15	′′(ξ	′′(ξ	PROPN
ejde-761	370	16	)	)	PUNCT
ejde-761	370	17	,	,	PUNCT
ejde-761	370	18	∆(|uxx|muxx	∆(|uxx|muxx	PROPN
ejde-761	370	19	)	)	PUNCT
ejde-761	370	20	=	=	SYM
ejde-761	370	21	(	(	PUNCT
ejde-761	370	22	t	t	PROPN
ejde-761	370	23	−	−	PROPN
ejde-761	370	24	t)(−α−2β)(m+1)[|f	t)(−α−2β)(m+1)[|f	PROPN
ejde-761	370	25	′′(ξ)|mf	′′(ξ)|mf	PROPN
ejde-761	370	26	′′(ξ)]′′.	′′(ξ)]′′.	NOUN
ejde-761	370	27	substituting	substitute	VERB
ejde-761	370	28	these	these	PRON
ejde-761	370	29	into	into	ADP
ejde-761	370	30	the	the	DET
ejde-761	370	31	pde	pde	NOUN
ejde-761	370	32	and	and	CCONJ
ejde-761	370	33	equating	equate	VERB
ejde-761	370	34	the	the	DET
ejde-761	370	35	powers	power	NOUN
ejde-761	370	36	of	of	ADP
ejde-761	370	37	(	(	PUNCT
ejde-761	370	38	t	t	PROPN
ejde-761	370	39	−	−	PROPN
ejde-761	370	40	t	t	PROPN
ejde-761	370	41	)	)	PUNCT
ejde-761	370	42	,	,	PUNCT
ejde-761	370	43	so	so	SCONJ
ejde-761	370	44	that	that	SCONJ
ejde-761	370	45	we	we	PRON
ejde-761	370	46	derive	derive	VERB
ejde-761	370	47	a	a	DET
ejde-761	370	48	proper	proper	ADJ
ejde-761	370	49	equation	equation	NOUN
ejde-761	370	50	in	in	ADP
ejde-761	370	51	terms	term	NOUN
ejde-761	370	52	of	of	ADP
ejde-761	370	53	the	the	DET
ejde-761	370	54	single	single	ADJ
ejde-761	370	55	variable	variable	NOUN
ejde-761	370	56	ξ	ξ	PROPN
ejde-761	370	57	,	,	PUNCT
ejde-761	370	58	αf(ξ	αf(ξ	NUM
ejde-761	370	59	)	)	PUNCT
ejde-761	371	1	+	+	CCONJ
ejde-761	371	2	βξf	βξf	PRON
ejde-761	371	3	′(ξ	′(ξ	NOUN
ejde-761	371	4	)	)	PUNCT
ejde-761	371	5	=	=	PUNCT
ejde-761	372	1	[	[	X
ejde-761	372	2	|f	|f	X
ejde-761	372	3	′′(ξ)|mf	′′(ξ)|mf	PROPN
ejde-761	372	4	′′(ξ)]′′	′′(ξ)]′′	PROPN
ejde-761	372	5	+	+	CCONJ
ejde-761	372	6	|f(ξ)|p−1f(ξ	|f(ξ)|p−1f(ξ	NOUN
ejde-761	372	7	)	)	PUNCT
ejde-761	372	8	.	.	PUNCT
ejde-761	373	1	now	now	ADV
ejde-761	373	2	,	,	PUNCT
ejde-761	373	3	we	we	PRON
ejde-761	373	4	introduce	introduce	VERB
ejde-761	373	5	a	a	DET
ejde-761	373	6	resolution	resolution	NOUN
ejde-761	373	7	for	for	ADP
ejde-761	373	8	this	this	DET
ejde-761	373	9	equation	equation	NOUN
ejde-761	373	10	in	in	ADP
ejde-761	373	11	the	the	DET
ejde-761	373	12	proximity	proximity	NOUN
ejde-761	373	13	of	of	ADP
ejde-761	373	14	f	f	PROPN
ejde-761	373	15	→	→	SYM
ejde-761	373	16	0	0	NUM
ejde-761	373	17	+	+	ADJ
ejde-761	373	18	,	,	PUNCT
ejde-761	373	19	hence	hence	ADV
ejde-761	373	20	let	let	VERB
ejde-761	373	21	us	we	PRON
ejde-761	373	22	consider	consider	VERB
ejde-761	373	23	the	the	DET
ejde-761	373	24	expression	expression	NOUN
ejde-761	373	25	βξf	βξf	X
ejde-761	373	26	′(ξ	′(ξ	NOUN
ejde-761	373	27	)	)	PUNCT
ejde-761	373	28	=	=	PUNCT
ejde-761	374	1	[	[	X
ejde-761	374	2	|f	|f	X
ejde-761	374	3	′′(ξ)|m	′′(ξ)|m	PROPN
ejde-761	374	4	f	f	PROPN
ejde-761	374	5	′′(ξ)]′′.	′′(ξ)]′′.	NOUN
ejde-761	374	6	this	this	DET
ejde-761	374	7	equation	equation	NOUN
ejde-761	374	8	was	be	AUX
ejde-761	374	9	solved	solve	VERB
ejde-761	374	10	based	base	VERB
ejde-761	374	11	on	on	ADP
ejde-761	374	12	a	a	DET
ejde-761	374	13	numerical	numerical	ADJ
ejde-761	374	14	procedure	procedure	NOUN
ejde-761	374	15	using	use	VERB
ejde-761	374	16	finite	finite	ADJ
ejde-761	374	17	difference	difference	NOUN
ejde-761	374	18	methods	method	NOUN
ejde-761	374	19	and	and	CCONJ
ejde-761	374	20	for	for	ADP
ejde-761	374	21	m	m	PROPN
ejde-761	374	22	=	=	SYM
ejde-761	374	23	3	3	NUM
ejde-761	374	24	and	and	CCONJ
ejde-761	374	25	p	p	NOUN
ejde-761	374	26	=	=	ADJ
ejde-761	374	27	2	2	X
ejde-761	374	28	.	.	PUNCT
ejde-761	375	1	the	the	DET
ejde-761	375	2	domain	domain	NOUN
ejde-761	375	3	[	[	X
ejde-761	375	4	0	0	NUM
ejde-761	375	5	,	,	PUNCT
ejde-761	375	6	ξmax	ξmax	ADV
ejde-761	375	7	]	]	PUNCT
ejde-761	375	8	was	be	AUX
ejde-761	375	9	discretized	discretize	VERB
ejde-761	375	10	into	into	ADP
ejde-761	375	11	n	n	NOUN
ejde-761	375	12	=	=	SYM
ejde-761	375	13	100	100	NUM
ejde-761	375	14	points	point	NOUN
ejde-761	375	15	,	,	PUNCT
ejde-761	375	16	and	and	CCONJ
ejde-761	375	17	central	central	ADJ
ejde-761	375	18	finite	finite	ADJ
ejde-761	375	19	differences	difference	NOUN
ejde-761	375	20	were	be	AUX
ejde-761	375	21	employed	employ	VERB
ejde-761	375	22	to	to	PART
ejde-761	375	23	approximate	approximate	VERB
ejde-761	375	24	the	the	DET
ejde-761	375	25	first	first	ADJ
ejde-761	375	26	,	,	PUNCT
ejde-761	375	27	second	second	ADJ
ejde-761	375	28	,	,	PUNCT
ejde-761	375	29	and	and	CCONJ
ejde-761	375	30	fourth	fourth	ADJ
ejde-761	375	31	derivatives	derivative	NOUN
ejde-761	375	32	of	of	ADP
ejde-761	375	33	the	the	DET
ejde-761	375	34	function	function	NOUN
ejde-761	375	35	f(ξ	f(ξ	NOUN
ejde-761	375	36	)	)	PUNCT
ejde-761	375	37	.	.	PUNCT
ejde-761	376	1	an	an	DET
ejde-761	376	2	initial	initial	ADJ
ejde-761	376	3	guess	guess	NOUN
ejde-761	376	4	,	,	PUNCT
ejde-761	376	5	of	of	ADP
ejde-761	376	6	a	a	DET
ejde-761	376	7	gaussian	gaussian	ADJ
ejde-761	376	8	-	-	PUNCT
ejde-761	376	9	like	like	ADJ
ejde-761	376	10	function	function	NOUN
ejde-761	376	11	(	(	PUNCT
ejde-761	376	12	f0(ξ	f0(ξ	X
ejde-761	376	13	)	)	PUNCT
ejde-761	376	14	=	=	SYM
ejde-761	376	15	exp(−ξ2	exp(−ξ2	NOUN
ejde-761	376	16	)	)	PUNCT
ejde-761	376	17	)	)	PUNCT
ejde-761	376	18	,	,	PUNCT
ejde-761	376	19	was	be	AUX
ejde-761	376	20	used	use	VERB
ejde-761	376	21	to	to	PART
ejde-761	376	22	start	start	VERB
ejde-761	376	23	the	the	DET
ejde-761	376	24	iterative	iterative	NOUN
ejde-761	376	25	process	process	NOUN
ejde-761	376	26	.	.	PUNCT
ejde-761	377	1	the	the	DET
ejde-761	377	2	system	system	NOUN
ejde-761	377	3	of	of	ADP
ejde-761	377	4	nonlinear	nonlinear	ADJ
ejde-761	377	5	equations	equation	NOUN
ejde-761	377	6	was	be	AUX
ejde-761	377	7	then	then	ADV
ejde-761	377	8	solved	solve	VERB
ejde-761	377	9	using	use	VERB
ejde-761	377	10	the	the	DET
ejde-761	377	11	fsolve	fsolve	NOUN
ejde-761	377	12	function	function	VERB
ejde-761	377	13	from	from	ADP
ejde-761	377	14	the	the	DET
ejde-761	377	15	scipy.optimize	scipy.optimize	PROPN
ejde-761	377	16	library	library	NOUN
ejde-761	377	17	in	in	ADP
ejde-761	377	18	python	python	PROPN
ejde-761	377	19	.	.	PUNCT
ejde-761	378	1	the	the	DET
ejde-761	378	2	numerical	numerical	ADJ
ejde-761	378	3	solution	solution	NOUN
ejde-761	378	4	obtained	obtain	VERB
ejde-761	378	5	was	be	AUX
ejde-761	378	6	subsequently	subsequently	ADV
ejde-761	378	7	compared	compare	VERB
ejde-761	378	8	with	with	ADP
ejde-761	378	9	an	an	DET
ejde-761	378	10	assumed	assumed	ADJ
ejde-761	378	11	analytical	analytical	ADJ
ejde-761	378	12	solution	solution	NOUN
ejde-761	378	13	of	of	ADP
ejde-761	378	14	the	the	DET
ejde-761	378	15	form	form	NOUN
ejde-761	378	16	fanalytical	fanalytical	ADJ
ejde-761	378	17	=	=	PUNCT
ejde-761	378	18	ce−aξ	ce−aξ	NOUN
ejde-761	378	19	,	,	PUNCT
ejde-761	378	20	and	and	CCONJ
ejde-761	378	21	the	the	DET
ejde-761	378	22	parameters	parameter	NOUN
ejde-761	378	23	c	c	PROPN
ejde-761	378	24	and	and	CCONJ
ejde-761	378	25	a	a	PRON
ejde-761	378	26	were	be	AUX
ejde-761	378	27	optimized	optimize	VERB
ejde-761	378	28	to	to	PART
ejde-761	378	29	minimize	minimize	VERB
ejde-761	378	30	the	the	DET
ejde-761	378	31	mean	mean	ADJ
ejde-761	378	32	absolute	absolute	ADJ
ejde-761	378	33	error	error	NOUN
ejde-761	378	34	between	between	ADP
ejde-761	378	35	the	the	DET
ejde-761	378	36	numerical	numerical	ADJ
ejde-761	378	37	and	and	CCONJ
ejde-761	378	38	analytical	analytical	ADJ
ejde-761	378	39	solutions	solution	NOUN
ejde-761	378	40	,	,	PUNCT
ejde-761	378	41	leading	lead	VERB
ejde-761	378	42	to	to	ADP
ejde-761	378	43	the	the	DET
ejde-761	378	44	following	follow	VERB
ejde-761	378	45	fitted	fit	VERB
ejde-761	378	46	parameters	parameter	NOUN
ejde-761	378	47	,	,	PUNCT
ejde-761	378	48	c	c	PROPN
ejde-761	378	49	≈	≈	PROPN
ejde-761	378	50	1.208	1.208	NUM
ejde-761	378	51	,	,	PUNCT
ejde-761	378	52	a	a	DET
ejde-761	378	53	≈	≈	PROPN
ejde-761	378	54	1.258	1.258	NUM
ejde-761	378	55	.	.	PUNCT
ejde-761	379	1	to	to	PART
ejde-761	379	2	achieve	achieve	VERB
ejde-761	379	3	this	this	PRON
ejde-761	379	4	,	,	PUNCT
ejde-761	379	5	we	we	PRON
ejde-761	379	6	minimize	minimize	VERB
ejde-761	379	7	the	the	DET
ejde-761	379	8	difference	difference	NOUN
ejde-761	379	9	between	between	ADP
ejde-761	379	10	the	the	DET
ejde-761	379	11	numerical	numerical	ADJ
ejde-761	379	12	solution	solution	NOUN
ejde-761	379	13	and	and	CCONJ
ejde-761	379	14	the	the	DET
ejde-761	379	15	analytical	analytical	ADJ
ejde-761	379	16	form	form	NOUN
ejde-761	379	17	fanalytical	fanalytical	ADJ
ejde-761	379	18	=	=	NOUN
ejde-761	379	19	ce−aξ	ce−aξ	NOUN
ejde-761	379	20	by	by	ADP
ejde-761	379	21	solving	solve	VERB
ejde-761	379	22	the	the	DET
ejde-761	379	23	optimization	optimization	NOUN
ejde-761	379	24	problem	problem	NOUN
ejde-761	379	25	min	min	PROPN
ejde-761	379	26	c	c	NOUN
ejde-761	379	27	,	,	PUNCT
ejde-761	379	28	a	a	DET
ejde-761	379	29	n∑	n∑	NOUN
ejde-761	379	30	i=1	i=1	PROPN
ejde-761	379	31	(	(	PUNCT
ejde-761	379	32	fnumerical(ξi)−	fnumerical(ξi)−	NOUN
ejde-761	379	33	ce−aξi)2	ce−aξi)2	PROPN
ejde-761	379	34	.	.	PUNCT
ejde-761	380	1	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	380	2	instability	instability	NOUN
ejde-761	380	3	of	of	ADP
ejde-761	380	4	energy	energy	NOUN
ejde-761	380	5	solutions	solution	NOUN
ejde-761	380	6	19	19	NUM
ejde-761	380	7	the	the	DET
ejde-761	380	8	least	least	ADJ
ejde-761	380	9	squares	square	NOUN
ejde-761	380	10	method	method	NOUN
ejde-761	380	11	involves	involve	VERB
ejde-761	380	12	finding	find	VERB
ejde-761	380	13	the	the	DET
ejde-761	380	14	parameters	parameter	NOUN
ejde-761	380	15	c	c	PROPN
ejde-761	380	16	and	and	CCONJ
ejde-761	380	17	a	a	DET
ejde-761	380	18	that	that	PRON
ejde-761	380	19	minimize	minimize	VERB
ejde-761	380	20	the	the	DET
ejde-761	380	21	sum	sum	NOUN
ejde-761	380	22	of	of	ADP
ejde-761	380	23	the	the	DET
ejde-761	380	24	squared	square	VERB
ejde-761	380	25	differences	difference	NOUN
ejde-761	380	26	between	between	ADP
ejde-761	380	27	the	the	DET
ejde-761	380	28	numerical	numerical	ADJ
ejde-761	380	29	solution	solution	NOUN
ejde-761	380	30	fnumerical(ξi	fnumerical(ξi	NOUN
ejde-761	380	31	)	)	PUNCT
ejde-761	380	32	and	and	CCONJ
ejde-761	380	33	the	the	DET
ejde-761	380	34	analytical	analytical	ADJ
ejde-761	380	35	solution	solution	NOUN
ejde-761	380	36	ce−aξi	ce−aξi	NOUN
ejde-761	380	37	over	over	ADP
ejde-761	380	38	all	all	DET
ejde-761	380	39	discretized	discretized	ADJ
ejde-761	380	40	points	point	NOUN
ejde-761	380	41	ξi	ξi	NOUN
ejde-761	380	42	.	.	PUNCT
ejde-761	381	1	to	to	PART
ejde-761	381	2	implement	implement	VERB
ejde-761	381	3	this	this	PRON
ejde-761	381	4	,	,	PUNCT
ejde-761	381	5	we	we	PRON
ejde-761	381	6	use	use	VERB
ejde-761	381	7	the	the	DET
ejde-761	381	8	curve	curve	NOUN
ejde-761	381	9	fit	fit	ADJ
ejde-761	381	10	function	function	NOUN
ejde-761	381	11	from	from	ADP
ejde-761	381	12	the	the	DET
ejde-761	381	13	scipy.optimize	scipy.optimize	PROPN
ejde-761	381	14	library	library	NOUN
ejde-761	381	15	in	in	ADP
ejde-761	381	16	python	python	PROPN
ejde-761	381	17	,	,	PUNCT
ejde-761	381	18	which	which	PRON
ejde-761	381	19	employs	employ	VERB
ejde-761	381	20	non	non	ADJ
ejde-761	381	21	-	-	ADJ
ejde-761	381	22	linear	linear	ADJ
ejde-761	381	23	least	least	ADJ
ejde-761	381	24	squares	square	NOUN
ejde-761	381	25	to	to	PART
ejde-761	381	26	fit	fit	VERB
ejde-761	381	27	the	the	DET
ejde-761	381	28	exponential	exponential	ADJ
ejde-761	381	29	model	model	NOUN
ejde-761	381	30	to	to	ADP
ejde-761	381	31	the	the	DET
ejde-761	381	32	numerical	numerical	PROPN
ejde-761	381	33	data	data	PROPN
ejde-761	381	34	.	.	PUNCT
ejde-761	382	1	the	the	DET
ejde-761	382	2	curve	curve	NOUN
ejde-761	382	3	fit	fit	ADJ
ejde-761	382	4	function	function	NOUN
ejde-761	382	5	returns	return	VERB
ejde-761	382	6	the	the	DET
ejde-761	382	7	optimal	optimal	ADJ
ejde-761	382	8	values	value	NOUN
ejde-761	382	9	of	of	ADP
ejde-761	382	10	c	c	PROPN
ejde-761	382	11	and	and	CCONJ
ejde-761	382	12	a	a	DET
ejde-761	382	13	that	that	PRON
ejde-761	382	14	minimize	minimize	VERB
ejde-761	382	15	the	the	DET
ejde-761	382	16	squared	square	VERB
ejde-761	382	17	error	error	NOUN
ejde-761	382	18	.	.	PUNCT
ejde-761	383	1	for	for	ADP
ejde-761	383	2	this	this	PRON
ejde-761	383	3	,	,	PUNCT
ejde-761	383	4	we	we	PRON
ejde-761	383	5	started	start	VERB
ejde-761	383	6	with	with	ADP
ejde-761	383	7	an	an	DET
ejde-761	383	8	initial	initial	ADJ
ejde-761	383	9	guess	guess	NOUN
ejde-761	383	10	for	for	ADP
ejde-761	383	11	the	the	DET
ejde-761	383	12	parameters	parameter	NOUN
ejde-761	383	13	c	c	NOUN
ejde-761	383	14	=	=	SYM
ejde-761	383	15	1	1	NUM
ejde-761	383	16	and	and	CCONJ
ejde-761	383	17	a	a	DET
ejde-761	383	18	=	=	ADJ
ejde-761	383	19	1	1	NUM
ejde-761	383	20	.	.	PUNCT
ejde-761	384	1	hence	hence	ADV
ejde-761	384	2	,	,	PUNCT
ejde-761	384	3	we	we	PRON
ejde-761	384	4	computed	compute	VERB
ejde-761	384	5	the	the	DET
ejde-761	384	6	error	error	NOUN
ejde-761	384	7	function	function	NOUN
ejde-761	384	8	that	that	PRON
ejde-761	384	9	was	be	AUX
ejde-761	384	10	defined	define	VERB
ejde-761	384	11	as	as	ADP
ejde-761	384	12	the	the	DET
ejde-761	384	13	sum	sum	NOUN
ejde-761	384	14	of	of	ADP
ejde-761	384	15	the	the	DET
ejde-761	384	16	squared	square	VERB
ejde-761	384	17	differences	difference	NOUN
ejde-761	384	18	between	between	ADP
ejde-761	384	19	the	the	DET
ejde-761	384	20	numerical	numerical	ADJ
ejde-761	384	21	solution	solution	NOUN
ejde-761	384	22	and	and	CCONJ
ejde-761	384	23	the	the	DET
ejde-761	384	24	analytical	analytical	ADJ
ejde-761	384	25	form	form	NOUN
ejde-761	384	26	.	.	PUNCT
ejde-761	385	1	this	this	DET
ejde-761	385	2	error	error	NOUN
ejde-761	385	3	function	function	NOUN
ejde-761	385	4	is	be	AUX
ejde-761	385	5	what	what	PRON
ejde-761	385	6	the	the	DET
ejde-761	385	7	optimization	optimization	NOUN
ejde-761	385	8	algorithm	algorithm	NOUN
ejde-761	385	9	seeks	seek	VERB
ejde-761	385	10	to	to	PART
ejde-761	385	11	minimize	minimize	VERB
ejde-761	385	12	.	.	PUNCT
ejde-761	386	1	particularly	particularly	ADV
ejde-761	386	2	,	,	PUNCT
ejde-761	386	3	the	the	DET
ejde-761	386	4	curve	curve	NOUN
ejde-761	386	5	fit	fit	ADJ
ejde-761	386	6	function	function	NOUN
ejde-761	386	7	applies	apply	VERB
ejde-761	386	8	an	an	DET
ejde-761	386	9	optimization	optimization	NOUN
ejde-761	386	10	algorithm	algorithm	NOUN
ejde-761	386	11	to	to	PART
ejde-761	386	12	adjust	adjust	VERB
ejde-761	386	13	the	the	DET
ejde-761	386	14	parameters	parameter	NOUN
ejde-761	386	15	c	c	PROPN
ejde-761	386	16	and	and	CCONJ
ejde-761	386	17	a	a	DET
ejde-761	386	18	iteratively	iteratively	ADV
ejde-761	386	19	.	.	PUNCT
ejde-761	387	1	in	in	ADP
ejde-761	387	2	each	each	DET
ejde-761	387	3	iteration	iteration	NOUN
ejde-761	387	4	,	,	PUNCT
ejde-761	387	5	the	the	DET
ejde-761	387	6	algorithm	algorithm	NOUN
ejde-761	387	7	evaluated	evaluate	VERB
ejde-761	387	8	the	the	DET
ejde-761	387	9	error	error	NOUN
ejde-761	387	10	function	function	NOUN
ejde-761	387	11	and	and	CCONJ
ejde-761	387	12	updated	update	VERB
ejde-761	387	13	the	the	DET
ejde-761	387	14	parameters	parameter	NOUN
ejde-761	387	15	to	to	PART
ejde-761	387	16	reduce	reduce	VERB
ejde-761	387	17	the	the	DET
ejde-761	387	18	error	error	NOUN
ejde-761	387	19	.	.	PUNCT
ejde-761	388	1	the	the	DET
ejde-761	388	2	process	process	NOUN
ejde-761	388	3	continued	continue	VERB
ejde-761	388	4	until	until	SCONJ
ejde-761	388	5	the	the	DET
ejde-761	388	6	algorithm	algorithm	NOUN
ejde-761	388	7	converges	converge	VERB
ejde-761	388	8	to	to	ADP
ejde-761	388	9	a	a	DET
ejde-761	388	10	solution	solution	NOUN
ejde-761	388	11	where	where	SCONJ
ejde-761	388	12	the	the	DET
ejde-761	388	13	parameters	parameter	NOUN
ejde-761	388	14	c	c	PROPN
ejde-761	388	15	and	and	CCONJ
ejde-761	388	16	a	a	DET
ejde-761	388	17	yield	yield	NOUN
ejde-761	388	18	the	the	DET
ejde-761	388	19	minimum	minimum	NOUN
ejde-761	388	20	error	error	NOUN
ejde-761	388	21	.	.	PUNCT
ejde-761	389	1	the	the	DET
ejde-761	389	2	resulting	result	VERB
ejde-761	389	3	fitted	fit	VERB
ejde-761	389	4	parameters	parameter	NOUN
ejde-761	389	5	provide	provide	VERB
ejde-761	389	6	an	an	DET
ejde-761	389	7	exponential	exponential	ADJ
ejde-761	389	8	function	function	NOUN
ejde-761	389	9	that	that	PRON
ejde-761	389	10	closely	closely	ADV
ejde-761	389	11	approximates	approximate	VERB
ejde-761	389	12	the	the	DET
ejde-761	389	13	numerical	numerical	ADJ
ejde-761	389	14	solution	solution	NOUN
ejde-761	389	15	,	,	PUNCT
ejde-761	389	16	as	as	SCONJ
ejde-761	389	17	illustrated	illustrate	VERB
ejde-761	389	18	in	in	ADP
ejde-761	389	19	the	the	DET
ejde-761	389	20	figure	figure	NOUN
ejde-761	389	21	5	5	NUM
ejde-761	389	22	.	.	PUNCT
ejde-761	390	1	the	the	DET
ejde-761	390	2	comparison	comparison	NOUN
ejde-761	390	3	demonstrates	demonstrate	VERB
ejde-761	390	4	that	that	SCONJ
ejde-761	390	5	the	the	DET
ejde-761	390	6	fitted	fit	VERB
ejde-761	390	7	analytical	analytical	ADJ
ejde-761	390	8	solution	solution	NOUN
ejde-761	390	9	matches	match	VERB
ejde-761	390	10	the	the	DET
ejde-761	390	11	numerical	numerical	ADJ
ejde-761	390	12	solution	solution	NOUN
ejde-761	390	13	with	with	ADP
ejde-761	390	14	a	a	DET
ejde-761	390	15	mean	mean	ADJ
ejde-761	390	16	absolute	absolute	ADJ
ejde-761	390	17	error	error	NOUN
ejde-761	390	18	of	of	ADP
ejde-761	390	19	approximately	approximately	ADV
ejde-761	390	20	0.0238	0.0238	NUM
ejde-761	390	21	,	,	PUNCT
ejde-761	390	22	indicating	indicate	VERB
ejde-761	390	23	a	a	DET
ejde-761	390	24	high	high	ADJ
ejde-761	390	25	degree	degree	NOUN
ejde-761	390	26	of	of	ADP
ejde-761	390	27	accuracy	accuracy	NOUN
ejde-761	390	28	.	.	PUNCT
ejde-761	391	1	figure	figure	NOUN
ejde-761	391	2	7	7	NUM
ejde-761	391	3	.	.	PUNCT
ejde-761	391	4	comparison	comparison	NOUN
ejde-761	391	5	of	of	ADP
ejde-761	391	6	numerical	numerical	ADJ
ejde-761	391	7	solution	solution	NOUN
ejde-761	391	8	and	and	CCONJ
ejde-761	391	9	fitted	fit	VERB
ejde-761	391	10	analytical	analytical	ADJ
ejde-761	391	11	solution	solution	NOUN
ejde-761	391	12	for	for	ADP
ejde-761	391	13	m	m	PROPN
ejde-761	391	14	=	=	SYM
ejde-761	391	15	3	3	NUM
ejde-761	391	16	and	and	CCONJ
ejde-761	391	17	p	p	NOUN
ejde-761	391	18	=	=	ADJ
ejde-761	391	19	2	2	X
ejde-761	391	20	.	.	PUNCT
ejde-761	392	1	the	the	DET
ejde-761	392	2	mean	mean	ADJ
ejde-761	392	3	absolute	absolute	ADJ
ejde-761	392	4	error	error	NOUN
ejde-761	392	5	between	between	ADP
ejde-761	392	6	for	for	ADP
ejde-761	392	7	solutions	solution	NOUN
ejde-761	392	8	is	be	AUX
ejde-761	392	9	of	of	ADP
ejde-761	392	10	approximately	approximately	ADV
ejde-761	392	11	0.0238	0.0238	NUM
ejde-761	392	12	showing	show	VERB
ejde-761	392	13	a	a	DET
ejde-761	392	14	high	high	ADJ
ejde-761	392	15	degree	degree	NOUN
ejde-761	392	16	of	of	ADP
ejde-761	392	17	accuracy	accuracy	NOUN
ejde-761	392	18	specially	specially	ADV
ejde-761	392	19	in	in	ADP
ejde-761	392	20	the	the	DET
ejde-761	392	21	asymptotic	asymptotic	NOUN
ejde-761	392	22	with	with	ADP
ejde-761	392	23	ξ	ξ	PROPN
ejde-761	392	24	≫	≫	PROPN
ejde-761	392	25	1	1	NUM
ejde-761	392	26	.	.	PUNCT
ejde-761	393	1	once	once	ADV
ejde-761	393	2	the	the	DET
ejde-761	393	3	self	self	NOUN
ejde-761	393	4	-	-	PUNCT
ejde-761	393	5	similar	similar	ADJ
ejde-761	393	6	profile	profile	NOUN
ejde-761	393	7	f(ξ	f(ξ	NOUN
ejde-761	393	8	)	)	PUNCT
ejde-761	393	9	is	be	AUX
ejde-761	393	10	obtained	obtain	VERB
ejde-761	393	11	numerically	numerically	ADV
ejde-761	393	12	,	,	PUNCT
ejde-761	393	13	we	we	PRON
ejde-761	393	14	can	can	AUX
ejde-761	393	15	reconstruct	reconstruct	VERB
ejde-761	393	16	the	the	DET
ejde-761	393	17	original	original	ADJ
ejde-761	393	18	solution	solution	NOUN
ejde-761	393	19	u(x	u(x	NOUN
ejde-761	393	20	,	,	PUNCT
ejde-761	393	21	t	t	NOUN
ejde-761	393	22	)	)	PUNCT
ejde-761	393	23	using	use	VERB
ejde-761	393	24	the	the	DET
ejde-761	393	25	self	self	NOUN
ejde-761	393	26	-	-	PUNCT
ejde-761	393	27	similar	similar	ADJ
ejde-761	393	28	form	form	NOUN
ejde-761	393	29	.	.	PUNCT
ejde-761	394	1	if	if	SCONJ
ejde-761	394	2	we	we	PRON
ejde-761	394	3	consider	consider	VERB
ejde-761	394	4	the	the	DET
ejde-761	394	5	cases	case	NOUN
ejde-761	394	6	ofm	ofm	PROPN
ejde-761	394	7	=	=	SYM
ejde-761	394	8	3	3	NUM
ejde-761	394	9	and	and	CCONJ
ejde-761	394	10	p	p	NOUN
ejde-761	394	11	=	=	PROPN
ejde-761	394	12	2	2	NUM
ejde-761	394	13	,	,	PUNCT
ejde-761	394	14	the	the	DET
ejde-761	394	15	solution	solution	NOUN
ejde-761	394	16	is	be	AUX
ejde-761	394	17	not	not	PART
ejde-761	394	18	expected	expect	VERB
ejde-761	394	19	to	to	PART
ejde-761	394	20	blow	blow	VERB
ejde-761	394	21	-	-	PUNCT
ejde-761	394	22	up	up	NOUN
ejde-761	394	23	in	in	ADP
ejde-761	394	24	finite	finite	ADJ
ejde-761	394	25	time	time	NOUN
ejde-761	394	26	as	as	ADP
ejde-761	394	27	α	α	NOUN
ejde-761	394	28	=	=	SYM
ejde-761	394	29	−1	−1	NOUN
ejde-761	394	30	.	.	PUNCT
ejde-761	395	1	to	to	PART
ejde-761	395	2	illustrate	illustrate	VERB
ejde-761	395	3	this	this	DET
ejde-761	395	4	process	process	NOUN
ejde-761	395	5	,	,	PUNCT
ejde-761	395	6	we	we	PRON
ejde-761	395	7	assume	assume	VERB
ejde-761	395	8	for	for	ADP
ejde-761	395	9	simplicity	simplicity	NOUN
ejde-761	395	10	that	that	PRON
ejde-761	395	11	t	t	NOUN
ejde-761	395	12	=	=	SYM
ejde-761	395	13	1	1	X
ejde-761	395	14	.	.	PUNCT
ejde-761	396	1	using	use	VERB
ejde-761	396	2	the	the	DET
ejde-761	396	3	numerical	numerical	ADJ
ejde-761	396	4	solution	solution	NOUN
ejde-761	396	5	for	for	ADP
ejde-761	396	6	f(ξ	f(ξ	NOUN
ejde-761	396	7	)	)	PUNCT
ejde-761	396	8	,	,	PUNCT
ejde-761	396	9	shown	show	VERB
ejde-761	396	10	in	in	ADP
ejde-761	396	11	figure	figure	NOUN
ejde-761	396	12	5	5	NUM
ejde-761	396	13	,	,	PUNCT
ejde-761	396	14	we	we	PRON
ejde-761	396	15	can	can	AUX
ejde-761	396	16	reconstruct	reconstruct	VERB
ejde-761	396	17	the	the	DET
ejde-761	396	18	original	original	ADJ
ejde-761	396	19	solution	solution	NOUN
ejde-761	396	20	u(x	u(x	NOUN
ejde-761	396	21	,	,	PUNCT
ejde-761	396	22	t	t	PROPN
ejde-761	396	23	)	)	PUNCT
ejde-761	396	24	as	as	SCONJ
ejde-761	396	25	provided	provide	VERB
ejde-761	396	26	in	in	ADP
ejde-761	396	27	figure	figure	NOUN
ejde-761	396	28	5	5	NUM
ejde-761	396	29	.	.	SYM
ejde-761	396	30	20	20	NUM
ejde-761	396	31	j.	j.	PROPN
ejde-761	396	32	l.	l.	PROPN
ejde-761	396	33	díaz	díaz	PROPN
ejde-761	396	34	palencia	palencia	PROPN
ejde-761	396	35	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	396	36	figure	figure	NOUN
ejde-761	396	37	8	8	NUM
ejde-761	396	38	.	.	PUNCT
ejde-761	397	1	reconstructed	reconstruct	VERB
ejde-761	397	2	solution	solution	NOUN
ejde-761	397	3	u(x	u(x	NOUN
ejde-761	397	4	,	,	PUNCT
ejde-761	397	5	t	t	PROPN
ejde-761	397	6	)	)	PUNCT
ejde-761	397	7	for	for	ADP
ejde-761	397	8	different	different	ADJ
ejde-761	397	9	times	time	NOUN
ejde-761	397	10	6	6	NUM
ejde-761	397	11	.	.	PUNCT
ejde-761	398	1	conclusions	conclusion	NOUN
ejde-761	398	2	the	the	DET
ejde-761	398	3	analysis	analysis	NOUN
ejde-761	398	4	followed	follow	VERB
ejde-761	398	5	in	in	ADP
ejde-761	398	6	the	the	DET
ejde-761	398	7	presented	present	VERB
ejde-761	398	8	study	study	NOUN
ejde-761	398	9	has	have	AUX
ejde-761	398	10	permitted	permit	VERB
ejde-761	398	11	to	to	PART
ejde-761	398	12	introduce	introduce	VERB
ejde-761	398	13	results	result	NOUN
ejde-761	398	14	on	on	ADP
ejde-761	398	15	instabilities	instability	NOUN
ejde-761	398	16	of	of	ADP
ejde-761	398	17	tws	tws	NOUN
ejde-761	398	18	for	for	ADP
ejde-761	398	19	a	a	DET
ejde-761	398	20	kind	kind	NOUN
ejde-761	398	21	of	of	ADP
ejde-761	398	22	problem	problem	NOUN
ejde-761	398	23	with	with	ADP
ejde-761	398	24	a	a	DET
ejde-761	398	25	super	super	ADJ
ejde-761	398	26	-	-	ADJ
ejde-761	398	27	linear	linear	ADJ
ejde-761	398	28	reaction	reaction	NOUN
ejde-761	398	29	and	and	CCONJ
ejde-761	398	30	higher	high	ADJ
ejde-761	398	31	order	order	NOUN
ejde-761	398	32	p	p	X
ejde-761	398	33	-	-	PUNCT
ejde-761	398	34	laplacian	laplacian	ADJ
ejde-761	398	35	operator	operator	NOUN
ejde-761	398	36	.	.	PUNCT
ejde-761	399	1	the	the	DET
ejde-761	399	2	analysis	analysis	NOUN
ejde-761	399	3	started	start	VERB
ejde-761	399	4	by	by	ADP
ejde-761	399	5	the	the	DET
ejde-761	399	6	introduction	introduction	NOUN
ejde-761	399	7	of	of	ADP
ejde-761	399	8	appropriate	appropriate	ADJ
ejde-761	399	9	functional	functional	ADJ
ejde-761	399	10	spaces	space	NOUN
ejde-761	399	11	to	to	PART
ejde-761	399	12	account	account	VERB
ejde-761	399	13	for	for	ADP
ejde-761	399	14	the	the	DET
ejde-761	399	15	particular	particular	ADJ
ejde-761	399	16	oscillating	oscillate	VERB
ejde-761	399	17	behaviour	behaviour	NOUN
ejde-761	399	18	of	of	ADP
ejde-761	399	19	solutions	solution	NOUN
ejde-761	399	20	along	along	ADP
ejde-761	399	21	with	with	ADP
ejde-761	399	22	compact	compact	ADJ
ejde-761	399	23	support	support	NOUN
ejde-761	399	24	properties	property	NOUN
ejde-761	399	25	.	.	PUNCT
ejde-761	400	1	the	the	DET
ejde-761	400	2	regularity	regularity	NOUN
ejde-761	400	3	of	of	ADP
ejde-761	400	4	the	the	DET
ejde-761	400	5	involved	involved	ADJ
ejde-761	400	6	operator	operator	NOUN
ejde-761	400	7	has	have	AUX
ejde-761	400	8	been	be	AUX
ejde-761	400	9	shown	show	VERB
ejde-761	400	10	in	in	ADP
ejde-761	400	11	the	the	DET
ejde-761	400	12	defined	define	VERB
ejde-761	400	13	functional	functional	ADJ
ejde-761	400	14	spaces	space	NOUN
ejde-761	400	15	concluding	conclude	VERB
ejde-761	400	16	that	that	SCONJ
ejde-761	400	17	any	any	DET
ejde-761	400	18	oscillating	oscillate	VERB
ejde-761	400	19	solution	solution	NOUN
ejde-761	400	20	(	(	PUNCT
ejde-761	400	21	in	in	ADP
ejde-761	400	22	the	the	DET
ejde-761	400	23	normhθ	normhθ	NOUN
ejde-761	400	24	)	)	PUNCT
ejde-761	400	25	can	can	AUX
ejde-761	400	26	be	be	AUX
ejde-761	400	27	bounded	bound	VERB
ejde-761	400	28	by	by	ADP
ejde-761	400	29	mollified	mollify	VERB
ejde-761	400	30	solutions	solution	NOUN
ejde-761	400	31	(	(	PUNCT
ejde-761	400	32	through	through	ADP
ejde-761	400	33	the	the	DET
ejde-761	400	34	normhm	normhm	NOUN
ejde-761	400	35	ρ	ρ	PROPN
ejde-761	400	36	)	)	PUNCT
ejde-761	400	37	and	and	CCONJ
ejde-761	400	38	compact	compact	ADJ
ejde-761	400	39	support	support	NOUN
ejde-761	400	40	solutions	solution	NOUN
ejde-761	400	41	(	(	PUNCT
ejde-761	400	42	norm	norm	NOUN
ejde-761	400	43	hm	hm	INTJ
ejde-761	400	44	0	0	NUM
ejde-761	400	45	)	)	PUNCT
ejde-761	400	46	.	.	PUNCT
ejde-761	401	1	afterward	afterward	ADV
ejde-761	401	2	,	,	PUNCT
ejde-761	401	3	the	the	DET
ejde-761	401	4	boundedness	boundedness	NOUN
ejde-761	401	5	of	of	ADP
ejde-761	401	6	solutions	solution	NOUN
ejde-761	401	7	was	be	AUX
ejde-761	401	8	provided	provide	VERB
ejde-761	401	9	making	make	VERB
ejde-761	401	10	use	use	NOUN
ejde-761	401	11	of	of	ADP
ejde-761	401	12	the	the	DET
ejde-761	401	13	defined	define	VERB
ejde-761	401	14	norms	norm	NOUN
ejde-761	401	15	and	and	CCONJ
ejde-761	401	16	considering	consider	VERB
ejde-761	401	17	energy	energy	NOUN
ejde-761	401	18	solutions	solution	NOUN
ejde-761	401	19	.	.	PUNCT
ejde-761	402	1	once	once	ADV
ejde-761	402	2	the	the	DET
ejde-761	402	3	regularity	regularity	NOUN
ejde-761	402	4	results	result	NOUN
ejde-761	402	5	were	be	AUX
ejde-761	402	6	presented	present	VERB
ejde-761	402	7	,	,	PUNCT
ejde-761	402	8	the	the	DET
ejde-761	402	9	problem	problem	NOUN
ejde-761	402	10	(	(	PUNCT
ejde-761	402	11	1.1	1.1	NUM
ejde-761	402	12	)	)	PUNCT
ejde-761	402	13	was	be	AUX
ejde-761	402	14	analyzed	analyze	VERB
ejde-761	402	15	in	in	ADP
ejde-761	402	16	the	the	DET
ejde-761	402	17	tw	tw	NOUN
ejde-761	402	18	domain	domain	NOUN
ejde-761	402	19	making	make	VERB
ejde-761	402	20	use	use	NOUN
ejde-761	402	21	of	of	ADP
ejde-761	402	22	different	different	ADJ
ejde-761	402	23	lemmas	lemma	NOUN
ejde-761	402	24	to	to	PART
ejde-761	402	25	show	show	VERB
ejde-761	402	26	the	the	DET
ejde-761	402	27	permanent	permanent	ADJ
ejde-761	402	28	instabilities	instability	NOUN
ejde-761	402	29	of	of	ADP
ejde-761	402	30	solutions	solution	NOUN
ejde-761	402	31	.	.	PUNCT
ejde-761	403	1	in	in	ADP
ejde-761	403	2	the	the	DET
ejde-761	403	3	tws	tws	NOUN
ejde-761	403	4	analysis	analysis	NOUN
ejde-761	403	5	,	,	PUNCT
ejde-761	403	6	a	a	DET
ejde-761	403	7	special	special	ADJ
ejde-761	403	8	emphasis	emphasis	NOUN
ejde-761	403	9	was	be	AUX
ejde-761	403	10	set	set	VERB
ejde-761	403	11	in	in	ADP
ejde-761	403	12	the	the	DET
ejde-761	403	13	null	null	ADJ
ejde-761	403	14	critical	critical	ADJ
ejde-761	403	15	poin	poin	NOUN
ejde-761	403	16	.	.	PUNCT
ejde-761	404	1	a	a	DET
ejde-761	404	2	numerical	numerical	ADJ
ejde-761	404	3	assessment	assessment	NOUN
ejde-761	404	4	was	be	AUX
ejde-761	404	5	introduced	introduce	VERB
ejde-761	404	6	to	to	PART
ejde-761	404	7	provide	provide	VERB
ejde-761	404	8	the	the	DET
ejde-761	404	9	homotopy	homotopy	NOUN
ejde-761	404	10	graphs	graph	NOUN
ejde-761	404	11	for	for	ADP
ejde-761	404	12	a	a	DET
ejde-761	404	13	wide	wide	ADJ
ejde-761	404	14	interval	interval	NOUN
ejde-761	404	15	of	of	ADP
ejde-761	404	16	tw	tw	NOUN
ejde-761	404	17	-	-	NOUN
ejde-761	404	18	speeds	speed	NOUN
ejde-761	404	19	,	,	PUNCT
ejde-761	404	20	along	along	ADP
ejde-761	404	21	with	with	ADP
ejde-761	404	22	the	the	DET
ejde-761	404	23	solutions	solution	NOUN
ejde-761	404	24	exact	exact	ADJ
ejde-761	404	25	profiles	profile	NOUN
ejde-761	404	26	.	.	PUNCT
ejde-761	405	1	this	this	DET
ejde-761	405	2	approach	approach	NOUN
ejde-761	405	3	permits	permit	VERB
ejde-761	405	4	to	to	PART
ejde-761	405	5	validate	validate	VERB
ejde-761	405	6	our	our	PRON
ejde-761	405	7	analytical	analytical	ADJ
ejde-761	405	8	assessments	assessment	NOUN
ejde-761	405	9	and	and	CCONJ
ejde-761	405	10	our	our	PRON
ejde-761	405	11	first	first	ADJ
ejde-761	405	12	postulated	postulate	VERB
ejde-761	405	13	intuition	intuition	NOUN
ejde-761	405	14	:	:	PUNCT
ejde-761	405	15	this	this	PRON
ejde-761	405	16	is	be	AUX
ejde-761	405	17	that	that	SCONJ
ejde-761	405	18	the	the	DET
ejde-761	405	19	null	null	ADJ
ejde-761	405	20	solution	solution	NOUN
ejde-761	405	21	acted	act	VERB
ejde-761	405	22	as	as	ADP
ejde-761	405	23	an	an	DET
ejde-761	405	24	attractor	attractor	NOUN
ejde-761	405	25	to	to	ADP
ejde-761	405	26	any	any	DET
ejde-761	405	27	oscillating	oscillate	VERB
ejde-761	405	28	flow	flow	NOUN
ejde-761	405	29	,	,	PUNCT
ejde-761	405	30	leading	lead	VERB
ejde-761	405	31	potentially	potentially	ADV
ejde-761	405	32	to	to	PART
ejde-761	405	33	hinder	hinder	VERB
ejde-761	405	34	blow	blow	VERB
ejde-761	405	35	-	-	PUNCT
ejde-761	405	36	up	up	ADP
ejde-761	405	37	formation	formation	NOUN
ejde-761	405	38	.	.	PUNCT
ejde-761	406	1	eventually	eventually	ADV
ejde-761	406	2	,	,	PUNCT
ejde-761	406	3	we	we	PRON
ejde-761	406	4	introduced	introduce	VERB
ejde-761	406	5	the	the	DET
ejde-761	406	6	scaling	scaling	ADJ
ejde-761	406	7	invariant	invariant	ADJ
ejde-761	406	8	properties	property	NOUN
ejde-761	406	9	of	of	ADP
ejde-761	406	10	the	the	DET
ejde-761	406	11	equation	equation	NOUN
ejde-761	406	12	and	and	CCONJ
ejde-761	406	13	obtained	obtain	VERB
ejde-761	406	14	self	self	NOUN
ejde-761	406	15	-	-	PUNCT
ejde-761	406	16	similar	similar	ADJ
ejde-761	406	17	solutions	solution	NOUN
ejde-761	406	18	.	.	PUNCT
ejde-761	407	1	references	reference	NOUN
ejde-761	407	2	[	[	X
ejde-761	407	3	1	1	NUM
ejde-761	407	4	]	]	X
ejde-761	407	5	adams	adams	PROPN
ejde-761	407	6	,	,	PUNCT
ejde-761	407	7	r.	r.	PROPN
ejde-761	407	8	a.	a.	PROPN
ejde-761	407	9	;	;	PUNCT
ejde-761	407	10	anisotropic	anisotropic	NOUN
ejde-761	407	11	sobolev	sobolev	NOUN
ejde-761	407	12	inequalities	inequality	NOUN
ejde-761	407	13	.	.	PUNCT
ejde-761	408	1	casopis	casopis	VERB
ejde-761	408	2	pro	pro	ADJ
ejde-761	408	3	pestovan’i	pestovan’i	NOUN
ejde-761	408	4	matematiky	matematiky	VERB
ejde-761	408	5	.	.	PUNCT
ejde-761	409	1	113.3	113.3	NUM
ejde-761	409	2	(	(	PUNCT
ejde-761	409	3	1988	1988	NUM
ejde-761	409	4	):	):	PUNCT
ejde-761	409	5	267	267	NUM
ejde-761	409	6	-	-	SYM
ejde-761	409	7	279	279	NUM
ejde-761	409	8	.	.	PUNCT
ejde-761	410	1	http://eudml.org/doc/19616	http://eudml.org/doc/19616	X
ejde-761	411	1	[	[	X
ejde-761	411	2	2	2	NUM
ejde-761	411	3	]	]	X
ejde-761	411	4	ahn	ahn	PROPN
ejde-761	411	5	,	,	PUNCT
ejde-761	411	6	j.	j.	PROPN
ejde-761	411	7	;	;	PUNCT
ejde-761	411	8	yoon	yoon	PROPN
ejde-761	411	9	,	,	PUNCT
ejde-761	411	10	c.	c.	PROPN
ejde-761	411	11	;	;	PUNCT
ejde-761	411	12	global	global	ADJ
ejde-761	411	13	well	well	ADJ
ejde-761	411	14	-	-	PUNCT
ejde-761	411	15	posedness	posedness	NOUN
ejde-761	411	16	and	and	CCONJ
ejde-761	411	17	stability	stability	NOUN
ejde-761	411	18	of	of	ADP
ejde-761	411	19	constant	constant	ADJ
ejde-761	411	20	equilibria	equilibrium	NOUN
ejde-761	411	21	in	in	ADP
ejde-761	411	22	parabolic	parabolic	ADJ
ejde-761	411	23	–	–	PUNCT
ejde-761	411	24	elliptic	elliptic	ADJ
ejde-761	411	25	chemotaxis	chemotaxis	ADJ
ejde-761	411	26	system	system	NOUN
ejde-761	411	27	without	without	ADP
ejde-761	411	28	gradient	gradient	ADJ
ejde-761	411	29	sensing	sensing	NOUN
ejde-761	411	30	.	.	PUNCT
ejde-761	412	1	nonlinearity	nonlinearity	NOUN
ejde-761	412	2	,	,	PUNCT
ejde-761	412	3	32	32	NUM
ejde-761	412	4	(	(	PUNCT
ejde-761	412	5	2019	2019	NUM
ejde-761	412	6	)	)	PUNCT
ejde-761	412	7	,	,	PUNCT
ejde-761	412	8	1327	1327	NUM
ejde-761	412	9	-	-	SYM
ejde-761	412	10	1351	1351	NUM
ejde-761	412	11	.	.	PUNCT
ejde-761	413	1	[	[	X
ejde-761	413	2	3	3	NUM
ejde-761	413	3	]	]	X
ejde-761	413	4	alexander	alexander	PROPN
ejde-761	413	5	,	,	PUNCT
ejde-761	413	6	j.	j.	PROPN
ejde-761	413	7	;	;	PUNCT
ejde-761	413	8	gardner	gardner	PROPN
ejde-761	413	9	,	,	PUNCT
ejde-761	413	10	r.	r.	PROPN
ejde-761	413	11	;	;	PUNCT
ejde-761	413	12	jones	jones	PROPN
ejde-761	413	13	,	,	PUNCT
ejde-761	413	14	c.	c.	PROPN
ejde-761	413	15	;	;	PUNCT
ejde-761	413	16	a	a	DET
ejde-761	413	17	topological	topological	ADJ
ejde-761	413	18	invariant	invariant	ADJ
ejde-761	413	19	arising	arise	VERB
ejde-761	413	20	in	in	ADP
ejde-761	413	21	the	the	DET
ejde-761	413	22	stability	stability	NOUN
ejde-761	413	23	analysis	analysis	NOUN
ejde-761	413	24	of	of	ADP
ejde-761	413	25	travelling	travel	VERB
ejde-761	413	26	waves	wave	NOUN
ejde-761	413	27	.	.	PUNCT
ejde-761	414	1	j.	j.	PROPN
ejde-761	414	2	reine	reine	PROPN
ejde-761	414	3	angew	angew	PROPN
ejde-761	414	4	.	.	PUNCT
ejde-761	415	1	math	math	NOUN
ejde-761	415	2	.	.	PUNCT
ejde-761	416	1	410	410	NUM
ejde-761	416	2	(	(	PUNCT
ejde-761	416	3	1990	1990	NUM
ejde-761	416	4	)	)	PUNCT
ejde-761	416	5	,	,	PUNCT
ejde-761	416	6	167–212	167–212	NUM
ejde-761	416	7	.	.	PUNCT
ejde-761	417	1	doi	doi	PROPN
ejde-761	417	2	10.1515	10.1515	NUM
ejde-761	417	3	/	/	SYM
ejde-761	417	4	crll.1990.410.167	crll.1990.410.167	NOUN
ejde-761	417	5	[	[	X
ejde-761	417	6	4	4	NUM
ejde-761	417	7	]	]	X
ejde-761	417	8	aronson	aronson	PROPN
ejde-761	417	9	,	,	PUNCT
ejde-761	417	10	d.	d.	PROPN
ejde-761	417	11	;	;	PUNCT
ejde-761	417	12	density	density	NOUN
ejde-761	417	13	-	-	PUNCT
ejde-761	417	14	dependent	dependent	ADJ
ejde-761	417	15	interaction	interaction	NOUN
ejde-761	417	16	-	-	PUNCT
ejde-761	417	17	diffusion	diffusion	NOUN
ejde-761	417	18	systems	system	NOUN
ejde-761	417	19	.	.	PUNCT
ejde-761	418	1	proc	proc	NOUN
ejde-761	418	2	.	.	PUNCT
ejde-761	419	1	adv	adv	PROPN
ejde-761	419	2	.	.	PUNCT
ejde-761	419	3	seminar	seminar	NOUN
ejde-761	419	4	on	on	ADP
ejde-761	419	5	dynamics	dynamic	NOUN
ejde-761	419	6	and	and	CCONJ
ejde-761	419	7	modeling	modeling	NOUN
ejde-761	419	8	of	of	ADP
ejde-761	419	9	reactive	reactive	ADJ
ejde-761	419	10	system	system	NOUN
ejde-761	419	11	,	,	PUNCT
ejde-761	419	12	academic	academic	ADJ
ejde-761	419	13	press	press	NOUN
ejde-761	419	14	,	,	PUNCT
ejde-761	419	15	new	new	PROPN
ejde-761	419	16	york	york	PROPN
ejde-761	419	17	,	,	PUNCT
ejde-761	419	18	1980	1980	NUM
ejde-761	419	19	.	.	PUNCT
ejde-761	420	1	ejde-2024/68	ejde-2024/68	NOUN
ejde-761	420	2	instability	instability	NOUN
ejde-761	420	3	of	of	ADP
ejde-761	420	4	energy	energy	NOUN
ejde-761	420	5	solutions	solution	NOUN
ejde-761	420	6	21	21	NUM
ejde-761	420	7	[	[	X
ejde-761	420	8	5	5	NUM
ejde-761	420	9	]	]	X
ejde-761	420	10	aronson	aronson	PROPN
ejde-761	420	11	,	,	PUNCT
ejde-761	420	12	d.	d.	PROPN
ejde-761	420	13	;	;	PUNCT
ejde-761	420	14	weinberger	weinberger	PROPN
ejde-761	420	15	,	,	PUNCT
ejde-761	420	16	h.	h.	PROPN
ejde-761	420	17	;	;	PUNCT
ejde-761	420	18	nonlinear	nonlinear	ADJ
ejde-761	420	19	diffusion	diffusion	NOUN
ejde-761	420	20	in	in	ADP
ejde-761	420	21	population	population	NOUN
ejde-761	420	22	genetics	genetic	NOUN
ejde-761	420	23	,	,	PUNCT
ejde-761	420	24	combustion	combustion	NOUN
ejde-761	420	25	and	and	CCONJ
ejde-761	420	26	nerve	nerve	NOUN
ejde-761	420	27	propagation	propagation	NOUN
ejde-761	420	28	.	.	PUNCT
ejde-761	421	1	partial	partial	ADJ
ejde-761	421	2	differential	differential	ADJ
ejde-761	421	3	equations	equation	NOUN
ejde-761	421	4	and	and	CCONJ
ejde-761	421	5	related	related	ADJ
ejde-761	421	6	topics	topic	NOUN
ejde-761	421	7	.	.	PUNCT
ejde-761	422	1	(	(	PUNCT
ejde-761	422	2	1975	1975	NUM
ejde-761	422	3	)	)	PUNCT
ejde-761	422	4	pub	pub	NOUN
ejde-761	422	5	.	.	PUNCT
ejde-761	423	1	,	,	PUNCT
ejde-761	423	2	new	new	PROPN
ejde-761	423	3	york	york	PROPN
ejde-761	423	4	,	,	PUNCT
ejde-761	423	5	5–49	5–49	PROPN
ejde-761	423	6	.	.	PUNCT
ejde-761	424	1	[	[	X
ejde-761	424	2	6	6	NUM
ejde-761	424	3	]	]	X
ejde-761	424	4	aronson	aronson	PROPN
ejde-761	424	5	,	,	PUNCT
ejde-761	424	6	d.	d.	PROPN
ejde-761	424	7	;	;	PUNCT
ejde-761	424	8	weinberger	weinberger	PROPN
ejde-761	424	9	,	,	PUNCT
ejde-761	424	10	h.	h.	PROPN
ejde-761	424	11	;	;	PUNCT
ejde-761	424	12	multidimensional	multidimensional	ADJ
ejde-761	424	13	nonlinear	nonlinear	ADJ
ejde-761	424	14	diffusion	diffusion	NOUN
ejde-761	424	15	arising	arise	VERB
ejde-761	424	16	in	in	ADP
ejde-761	424	17	population	population	NOUN
ejde-761	424	18	genetics	genetic	NOUN
ejde-761	424	19	.	.	PUNCT
ejde-761	425	1	adv	adv	INTJ
ejde-761	425	2	.	.	PUNCT
ejde-761	426	1	in	in	ADP
ejde-761	426	2	math	math	NOUN
ejde-761	426	3	.	.	PUNCT
ejde-761	427	1	30	30	NUM
ejde-761	427	2	(	(	PUNCT
ejde-761	427	3	1978	1978	NUM
ejde-761	427	4	)	)	PUNCT
ejde-761	427	5	,	,	PUNCT
ejde-761	427	6	33–76	33–76	NUM
ejde-761	427	7	.	.	PUNCT
ejde-761	428	1	[	[	X
ejde-761	428	2	7	7	NUM
ejde-761	428	3	]	]	X
ejde-761	428	4	audrito	audrito	PROPN
ejde-761	428	5	,	,	PUNCT
ejde-761	428	6	a.	a.	NOUN
ejde-761	428	7	;	;	PUNCT
ejde-761	428	8	vázquez	vázquez	PROPN
ejde-761	428	9	,	,	PUNCT
ejde-761	428	10	j.	j.	PROPN
ejde-761	428	11	l.	l.	PROPN
ejde-761	428	12	;	;	PUNCT
ejde-761	428	13	the	the	DET
ejde-761	428	14	fisher	fisher	PROPN
ejde-761	428	15	–	–	PUNCT
ejde-761	428	16	kpp	kpp	PROPN
ejde-761	428	17	problem	problem	NOUN
ejde-761	428	18	with	with	ADP
ejde-761	428	19	doubly	doubly	ADV
ejde-761	428	20	nonlinear	nonlinear	ADJ
ejde-761	428	21	“	"	PUNCT
ejde-761	428	22	fast	fast	ADJ
ejde-761	428	23	”	"	PUNCT
ejde-761	428	24	diffusion	diffusion	NOUN
ejde-761	428	25	,	,	PUNCT
ejde-761	428	26	nonlinear	nonlinear	ADJ
ejde-761	428	27	analysis	analysis	NOUN
ejde-761	428	28	,	,	PUNCT
ejde-761	428	29	157	157	NUM
ejde-761	428	30	(	(	PUNCT
ejde-761	428	31	2017	2017	NUM
ejde-761	428	32	)	)	PUNCT
ejde-761	428	33	,	,	PUNCT
ejde-761	428	34	212	212	NUM
ejde-761	428	35	-	-	SYM
ejde-761	428	36	248	248	NUM
ejde-761	428	37	.	.	PUNCT
ejde-761	429	1	[	[	X
ejde-761	429	2	8	8	NUM
ejde-761	429	3	]	]	X
ejde-761	429	4	benedek	benedek	NOUN
ejde-761	429	5	,	,	PUNCT
ejde-761	429	6	a.	a.	NOUN
ejde-761	429	7	;	;	PUNCT
ejde-761	429	8	panzone	panzone	NOUN
ejde-761	429	9	,	,	PUNCT
ejde-761	429	10	r.	r.	PROPN
ejde-761	429	11	;	;	PUNCT
ejde-761	429	12	the	the	DET
ejde-761	429	13	spaces	space	NOUN
ejde-761	429	14	lp	lp	NOUN
ejde-761	429	15	with	with	ADP
ejde-761	429	16	mixed	mixed	ADJ
ejde-761	429	17	norm	norm	NOUN
ejde-761	429	18	.	.	PUNCT
ejde-761	430	1	duke	duke	PROPN
ejde-761	430	2	math	math	PROPN
ejde-761	430	3	.	.	PUNCT
ejde-761	431	1	j.	j.	PROPN
ejde-761	431	2	28	28	NUM
ejde-761	431	3	(	(	PUNCT
ejde-761	431	4	1961	1961	NUM
ejde-761	431	5	)	)	PUNCT
ejde-761	431	6	,	,	PUNCT
ejde-761	431	7	301	301	NUM
ejde-761	431	8	-	-	SYM
ejde-761	431	9	324	324	NUM
ejde-761	431	10	.	.	PUNCT
ejde-761	432	1	[	[	X
ejde-761	432	2	9	9	NUM
ejde-761	432	3	]	]	X
ejde-761	432	4	bhatti	bhatti	PROPN
ejde-761	432	5	,	,	PUNCT
ejde-761	432	6	m.	m.	NOUN
ejde-761	432	7	;	;	PUNCT
ejde-761	432	8	zeeshan	zeeshan	PROPN
ejde-761	432	9	,	,	PUNCT
ejde-761	432	10	a.	a.	NOUN
ejde-761	432	11	;	;	PUNCT
ejde-761	432	12	ellahi	ellahi	PROPN
ejde-761	432	13	,	,	PUNCT
ejde-761	432	14	r.	r.	PROPN
ejde-761	432	15	;	;	PUNCT
ejde-761	432	16	anwar	anwar	PROPN
ejde-761	432	17	b’eg	b’eg	PROPN
ejde-761	432	18	,	,	PUNCT
ejde-761	432	19	o.	o.	PROPN
ejde-761	432	20	;	;	PUNCT
ejde-761	432	21	kadir	kadir	PROPN
ejde-761	432	22	,	,	PUNCT
ejde-761	432	23	a.	a.	NOUN
ejde-761	432	24	;	;	PUNCT
ejde-761	432	25	effects	effect	NOUN
ejde-761	432	26	of	of	ADP
ejde-761	432	27	coagulation	coagulation	NOUN
ejde-761	432	28	on	on	ADP
ejde-761	432	29	the	the	DET
ejde-761	432	30	two	two	NUM
ejde-761	432	31	-	-	PUNCT
ejde-761	432	32	phase	phase	NOUN
ejde-761	432	33	peristaltic	peristaltic	ADJ
ejde-761	432	34	pumping	pumping	NOUN
ejde-761	432	35	of	of	ADP
ejde-761	432	36	magnetized	magnetized	ADJ
ejde-761	432	37	prandtl	prandtl	NOUN
ejde-761	432	38	biofluid	biofluid	NOUN
ejde-761	432	39	through	through	ADP
ejde-761	432	40	an	an	DET
ejde-761	432	41	endoscopic	endoscopic	ADJ
ejde-761	432	42	annular	annular	ADJ
ejde-761	432	43	geometry	geometry	NOUN
ejde-761	432	44	containing	contain	VERB
ejde-761	432	45	a	a	DET
ejde-761	432	46	porous	porous	ADJ
ejde-761	432	47	medium	medium	NOUN
ejde-761	432	48	,	,	PUNCT
ejde-761	432	49	chin	chin	NOUN
ejde-761	432	50	.	.	PUNCT
ejde-761	433	1	j.	j.	PROPN
ejde-761	433	2	phys	phys	PROPN
ejde-761	433	3	.	.	PUNCT
ejde-761	434	1	58	58	NUM
ejde-761	434	2	(	(	PUNCT
ejde-761	434	3	2019	2019	NUM
ejde-761	434	4	)	)	PUNCT
ejde-761	434	5	,	,	PUNCT
ejde-761	434	6	222	222	NUM
ejde-761	434	7	-	-	SYM
ejde-761	434	8	23	23	NUM
ejde-761	434	9	.	.	PUNCT
ejde-761	435	1	doi	doi	PROPN
ejde-761	435	2	10.1016	10.1016	NUM
ejde-761	435	3	/	/	SYM
ejde-761	435	4	j.cjph.2019.02.004	j.cjph.2019.02.004	NOUN
ejde-761	435	5	.	.	PUNCT
ejde-761	436	1	[	[	X
ejde-761	436	2	10	10	NUM
ejde-761	436	3	]	]	SYM
ejde-761	436	4	bognar	bognar	NOUN
ejde-761	436	5	,	,	PUNCT
ejde-761	436	6	g.	g.	PROPN
ejde-761	436	7	;	;	PUNCT
ejde-761	436	8	numerical	numerical	ADJ
ejde-761	436	9	and	and	CCONJ
ejde-761	436	10	analytic	analytic	ADJ
ejde-761	436	11	investigation	investigation	NOUN
ejde-761	436	12	of	of	ADP
ejde-761	436	13	some	some	DET
ejde-761	436	14	nonlinear	nonlinear	ADJ
ejde-761	436	15	problems	problem	NOUN
ejde-761	436	16	in	in	ADP
ejde-761	436	17	fluid	fluid	ADJ
ejde-761	436	18	mechanics	mechanic	NOUN
ejde-761	436	19	.	.	PUNCT
ejde-761	437	1	comp	comp	PROPN
ejde-761	437	2	.	.	PUNCT
ejde-761	438	1	and	and	CCONJ
ejde-761	438	2	sim	sim	ADJ
ejde-761	438	3	.	.	PUNCT
ejde-761	439	1	in	in	ADP
ejde-761	439	2	modern	modern	ADJ
ejde-761	439	3	sci	sci	PROPN
ejde-761	439	4	.	.	PUNCT
ejde-761	439	5	vol.ii	vol.ii	PROPN
ejde-761	439	6	(	(	PUNCT
ejde-761	439	7	2008	2008	NUM
ejde-761	439	8	)	)	PUNCT
ejde-761	439	9	,	,	PUNCT
ejde-761	439	10	pp.172	pp.172	PROPN
ejde-761	439	11	-	-	NOUN
ejde-761	439	12	179	179	NUM
ejde-761	439	13	.	.	PUNCT
ejde-761	440	1	[	[	X
ejde-761	440	2	11	11	NUM
ejde-761	440	3	]	]	SYM
ejde-761	440	4	bonheure	bonheure	NOUN
ejde-761	440	5	,	,	PUNCT
ejde-761	440	6	d.	d.	PROPN
ejde-761	440	7	;	;	PUNCT
ejde-761	440	8	sánchez	sánchez	PROPN
ejde-761	440	9	,	,	PUNCT
ejde-761	440	10	l.	l.	PROPN
ejde-761	440	11	;	;	PUNCT
ejde-761	440	12	heteroclinics	heteroclinic	NOUN
ejde-761	440	13	orbits	orbit	NOUN
ejde-761	440	14	for	for	ADP
ejde-761	440	15	some	some	DET
ejde-761	440	16	classes	class	NOUN
ejde-761	440	17	of	of	ADP
ejde-761	440	18	second	second	ADJ
ejde-761	440	19	and	and	CCONJ
ejde-761	440	20	fourth	fourth	ADJ
ejde-761	440	21	order	order	NOUN
ejde-761	440	22	differential	differential	ADJ
ejde-761	440	23	equations	equation	NOUN
ejde-761	440	24	.	.	PUNCT
ejde-761	441	1	handbook	handbook	NOUN
ejde-761	441	2	of	of	ADP
ejde-761	441	3	differential	differential	ADJ
ejde-761	441	4	equations	equation	NOUN
ejde-761	441	5	.	.	PUNCT
ejde-761	442	1	3(06	3(06	NUM
ejde-761	442	2	)	)	PUNCT
ejde-761	442	3	(	(	PUNCT
ejde-761	442	4	2006	2006	NUM
ejde-761	442	5	)	)	PUNCT
ejde-761	442	6	,	,	PUNCT
ejde-761	442	7	103	103	NUM
ejde-761	442	8	-	-	SYM
ejde-761	442	9	202	202	NUM
ejde-761	442	10	.	.	PUNCT
ejde-761	443	1	[	[	X
ejde-761	443	2	12	12	NUM
ejde-761	443	3	]	]	PUNCT
ejde-761	443	4	carelman	carelman	NOUN
ejde-761	443	5	,	,	PUNCT
ejde-761	443	6	t.	t.	PROPN
ejde-761	443	7	;	;	PUNCT
ejde-761	443	8	problemes	probleme	NOUN
ejde-761	443	9	mathematiques	mathematique	NOUN
ejde-761	443	10	dans	dans	PROPN
ejde-761	443	11	la	la	PROPN
ejde-761	443	12	theorie	theorie	PROPN
ejde-761	443	13	cinetique	cinetique	PROPN
ejde-761	443	14	de	de	X
ejde-761	443	15	gas	gas	NOUN
ejde-761	443	16	,	,	PUNCT
ejde-761	443	17	almquistwiksells	almquistwiksell	NOUN
ejde-761	443	18	,	,	PUNCT
ejde-761	443	19	uppsala	uppsala	PROPN
ejde-761	443	20	,	,	PUNCT
ejde-761	443	21	1957	1957	NUM
ejde-761	443	22	.	.	PUNCT
ejde-761	444	1	[	[	X
ejde-761	444	2	13	13	NUM
ejde-761	444	3	]	]	X
ejde-761	444	4	cho	cho	PROPN
ejde-761	444	5	,	,	PUNCT
ejde-761	444	6	e.	e.	PROPN
ejde-761	444	7	;	;	PUNCT
ejde-761	444	8	kim	kim	PROPN
ejde-761	444	9	,	,	PUNCT
ejde-761	444	10	y.	y.	PROPN
ejde-761	444	11	j.	j.	PROPN
ejde-761	444	12	;	;	PUNCT
ejde-761	444	13	starvation	starvation	NOUN
ejde-761	444	14	driven	drive	VERB
ejde-761	444	15	diffusion	diffusion	NOUN
ejde-761	444	16	as	as	ADP
ejde-761	444	17	a	a	DET
ejde-761	444	18	survival	survival	NOUN
ejde-761	444	19	strategy	strategy	NOUN
ejde-761	444	20	of	of	ADP
ejde-761	444	21	biological	biological	ADJ
ejde-761	444	22	organisms	organism	NOUN
ejde-761	444	23	bull	bull	NOUN
ejde-761	444	24	.	.	PUNCT
ejde-761	445	1	math	math	NOUN
ejde-761	445	2	.	.	PUNCT
ejde-761	446	1	biol	biol	PROPN
ejde-761	446	2	.	.	PUNCT
ejde-761	446	3	,	,	PUNCT
ejde-761	446	4	75	75	NUM
ejde-761	446	5	(	(	PUNCT
ejde-761	446	6	2013	2013	NUM
ejde-761	446	7	)	)	PUNCT
ejde-761	446	8	,	,	PUNCT
ejde-761	446	9	845	845	NUM
ejde-761	446	10	-	-	SYM
ejde-761	446	11	870	870	NUM
ejde-761	446	12	.	.	PUNCT
ejde-761	447	1	[	[	X
ejde-761	447	2	14	14	NUM
ejde-761	447	3	]	]	X
ejde-761	447	4	cohen	cohen	PROPN
ejde-761	447	5	,	,	PUNCT
ejde-761	447	6	d.	d.	PROPN
ejde-761	447	7	s.	s.	PROPN
ejde-761	447	8	;	;	PUNCT
ejde-761	447	9	murray	murray	PROPN
ejde-761	447	10	,	,	PUNCT
ejde-761	447	11	j.	j.	PROPN
ejde-761	447	12	d.	d.	PROPN
ejde-761	447	13	;	;	PUNCT
ejde-761	447	14	a	a	DET
ejde-761	447	15	generalized	generalize	VERB
ejde-761	447	16	diffusion	diffusion	NOUN
ejde-761	447	17	model	model	NOUN
ejde-761	447	18	for	for	ADP
ejde-761	447	19	growth	growth	NOUN
ejde-761	447	20	and	and	CCONJ
ejde-761	447	21	dispersal	dispersal	NOUN
ejde-761	447	22	in	in	ADP
ejde-761	447	23	a	a	DET
ejde-761	447	24	population	population	NOUN
ejde-761	447	25	.	.	PUNCT
ejde-761	448	1	j.	j.	PROPN
ejde-761	448	2	math	math	PROPN
ejde-761	448	3	.	.	PUNCT
ejde-761	449	1	biology	biology	NOUN
ejde-761	449	2	12	12	NUM
ejde-761	449	3	(	(	PUNCT
ejde-761	449	4	1981	1981	NUM
ejde-761	449	5	)	)	PUNCT
ejde-761	449	6	,	,	PUNCT
ejde-761	449	7	237–249	237–249	NUM
ejde-761	449	8	.	.	PUNCT
ejde-761	450	1	doi	doi	PROPN
ejde-761	450	2	10.1007	10.1007	NUM
ejde-761	450	3	/	/	SYM
ejde-761	450	4	bf00276132	bf00276132	PROPN
ejde-761	451	1	[	[	X
ejde-761	451	2	15	15	NUM
ejde-761	451	3	]	]	X
ejde-761	451	4	coutsias	coutsias	PROPN
ejde-761	451	5	,	,	PUNCT
ejde-761	451	6	e.	e.	PROPN
ejde-761	451	7	a.	a.	PROPN
ejde-761	451	8	;	;	PUNCT
ejde-761	451	9	some	some	DET
ejde-761	451	10	effects	effect	NOUN
ejde-761	451	11	of	of	ADP
ejde-761	451	12	spatial	spatial	ADJ
ejde-761	451	13	nonuniformities	nonuniformity	NOUN
ejde-761	451	14	in	in	ADP
ejde-761	451	15	chemically	chemically	ADV
ejde-761	451	16	reacting	react	VERB
ejde-761	451	17	systems	system	NOUN
ejde-761	451	18	.	.	PUNCT
ejde-761	452	1	california	california	PROPN
ejde-761	452	2	institute	institute	PROPN
ejde-761	452	3	of	of	ADP
ejde-761	452	4	technology	technology	PROPN
ejde-761	452	5	,	,	PUNCT
ejde-761	452	6	1980	1980	NUM
ejde-761	452	7	.	.	PUNCT
ejde-761	453	1	[	[	X
ejde-761	453	2	16	16	NUM
ejde-761	453	3	]	]	X
ejde-761	453	4	dee	dee	PROPN
ejde-761	453	5	,	,	PUNCT
ejde-761	453	6	g.	g.	PROPN
ejde-761	453	7	t.	t.	PROPN
ejde-761	453	8	;	;	PUNCT
ejde-761	453	9	van	van	PROPN
ejde-761	453	10	sarloos	sarloos	PROPN
ejde-761	453	11	,	,	PUNCT
ejde-761	453	12	w.	w.	NOUN
ejde-761	453	13	;	;	PUNCT
ejde-761	453	14	bistable	bistable	ADJ
ejde-761	453	15	systems	system	NOUN
ejde-761	453	16	with	with	ADP
ejde-761	453	17	propagating	propagate	VERB
ejde-761	453	18	fronts	front	NOUN
ejde-761	453	19	leading	lead	VERB
ejde-761	453	20	to	to	ADP
ejde-761	453	21	pattern	pattern	NOUN
ejde-761	453	22	formation	formation	NOUN
ejde-761	453	23	.	.	PUNCT
ejde-761	454	1	physical	physical	ADJ
ejde-761	454	2	review	review	NOUN
ejde-761	454	3	letter	letter	NOUN
ejde-761	454	4	,	,	PUNCT
ejde-761	454	5	volume	volume	NOUN
ejde-761	454	6	60	60	NUM
ejde-761	454	7	,	,	PUNCT
ejde-761	454	8	1998	1998	NUM
ejde-761	454	9	.	.	PUNCT
ejde-761	455	1	[	[	X
ejde-761	455	2	17	17	NUM
ejde-761	455	3	]	]	SYM
ejde-761	455	4	dı́az	dı́az	PROPN
ejde-761	455	5	,	,	PUNCT
ejde-761	455	6	j.	j.	PROPN
ejde-761	455	7	l.	l.	PROPN
ejde-761	455	8	;	;	PUNCT
ejde-761	455	9	(	(	PUNCT
ejde-761	455	10	2022	2022	NUM
ejde-761	455	11	)	)	PUNCT
ejde-761	455	12	non	non	ADJ
ejde-761	455	13	-	-	ADJ
ejde-761	455	14	lipschitz	lipschitz	ADJ
ejde-761	455	15	heterogeneous	heterogeneous	ADJ
ejde-761	455	16	reaction	reaction	NOUN
ejde-761	455	17	with	with	ADP
ejde-761	455	18	a	a	DET
ejde-761	455	19	p	p	ADJ
ejde-761	455	20	-	-	PUNCT
ejde-761	455	21	laplacian	laplacian	ADJ
ejde-761	455	22	operator	operator	NOUN
ejde-761	455	23	.	.	PUNCT
ejde-761	456	1	aims	aim	VERB
ejde-761	456	2	mathematics	mathematic	NOUN
ejde-761	456	3	,	,	PUNCT
ejde-761	456	4	7(3	7(3	NUM
ejde-761	456	5	)	)	PUNCT
ejde-761	456	6	(	(	PUNCT
ejde-761	456	7	2022	2022	NUM
ejde-761	456	8	):	):	PUNCT
ejde-761	456	9	3395	3395	NUM
ejde-761	456	10	-	-	PUNCT
ejde-761	456	11	3417	3417	NUM
ejde-761	456	12	.	.	PUNCT
ejde-761	457	1	doi	doi	NOUN
ejde-761	457	2	10.3934	10.3934	NUM
ejde-761	457	3	/	/	SYM
ejde-761	457	4	math.2022189	math.2022189	PROPN
ejde-761	458	1	[	[	SYM
ejde-761	458	2	18	18	NUM
ejde-761	458	3	]	]	X
ejde-761	458	4	durham	durham	PROPN
ejde-761	458	5	,	,	PUNCT
ejde-761	458	6	a.	a.	PROPN
ejde-761	458	7	c.	c.	PROPN
ejde-761	458	8	;	;	PUNCT
ejde-761	458	9	ridgway	ridgway	PROPN
ejde-761	458	10	,	,	PUNCT
ejde-761	458	11	e.	e.	PROPN
ejde-761	458	12	b.	b.	PROPN
ejde-761	458	13	;	;	PUNCT
ejde-761	458	14	control	control	NOUN
ejde-761	458	15	of	of	ADP
ejde-761	458	16	chemotaxis	chemotaxis	NOUN
ejde-761	458	17	in	in	ADP
ejde-761	458	18	physarum	physarum	PROPN
ejde-761	458	19	polycephalum	polycephalum	PROPN
ejde-761	458	20	.	.	PUNCT
ejde-761	459	1	j.	j.	PROPN
ejde-761	459	2	cell	cell	PROPN
ejde-761	459	3	.	.	PUNCT
ejde-761	460	1	biol	biol	PROPN
ejde-761	460	2	.	.	PUNCT
ejde-761	461	1	69	69	NUM
ejde-761	461	2	(	(	PUNCT
ejde-761	461	3	1976	1976	NUM
ejde-761	461	4	)	)	PUNCT
ejde-761	461	5	,	,	PUNCT
ejde-761	461	6	218–223	218–223	NUM
ejde-761	461	7	.	.	PUNCT
ejde-761	462	1	doi	doi	PROPN
ejde-761	462	2	10.1083	10.1083	NUM
ejde-761	462	3	/	/	SYM
ejde-761	462	4	jcb.69.1.218	jcb.69.1.218	NOUN
ejde-761	462	5	.	.	PUNCT
ejde-761	463	1	[	[	X
ejde-761	463	2	19	19	NUM
ejde-761	463	3	]	]	X
ejde-761	463	4	egorov	egorov	PROPN
ejde-761	463	5	,	,	PUNCT
ejde-761	463	6	y.	y.	PROPN
ejde-761	463	7	;	;	PUNCT
ejde-761	463	8	galaktionov	galaktionov	PROPN
ejde-761	463	9	,	,	PUNCT
ejde-761	463	10	v.	v.	PROPN
ejde-761	463	11	;	;	PUNCT
ejde-761	463	12	kondratiev	kondratiev	PROPN
ejde-761	463	13	,	,	PUNCT
ejde-761	463	14	v.	v.	PROPN
ejde-761	463	15	;	;	PUNCT
ejde-761	463	16	pohozaev	pohozaev	PROPN
ejde-761	463	17	,	,	PUNCT
ejde-761	463	18	s.	s.	PROPN
ejde-761	463	19	;	;	PUNCT
ejde-761	463	20	global	global	ADJ
ejde-761	463	21	solutions	solution	NOUN
ejde-761	463	22	of	of	ADP
ejde-761	463	23	higher	high	ADJ
ejde-761	463	24	-	-	PUNCT
ejde-761	463	25	order	order	NOUN
ejde-761	463	26	semilinear	semilinear	PROPN
ejde-761	463	27	parabolic	parabolic	NOUN
ejde-761	463	28	equations	equation	NOUN
ejde-761	463	29	in	in	ADP
ejde-761	463	30	the	the	DET
ejde-761	463	31	supercritical	supercritical	ADJ
ejde-761	463	32	range	range	NOUN
ejde-761	463	33	.	.	PUNCT
ejde-761	464	1	adv	adv	PROPN
ejde-761	464	2	.	.	PROPN
ejde-761	464	3	differ	differ	VERB
ejde-761	464	4	.	.	PUNCT
ejde-761	465	1	equat	equat	NOUN
ejde-761	465	2	.	.	PUNCT
ejde-761	466	1	9	9	NUM
ejde-761	466	2	(	(	PUNCT
ejde-761	466	3	2004	2004	NUM
ejde-761	466	4	)	)	PUNCT
ejde-761	466	5	,	,	PUNCT
ejde-761	466	6	10091038	10091038	NUM
ejde-761	466	7	.	.	PUNCT
ejde-761	467	1	[	[	X
ejde-761	467	2	20	20	NUM
ejde-761	467	3	]	]	SYM
ejde-761	467	4	ellahi	ellahi	PROPN
ejde-761	467	5	,	,	PUNCT
ejde-761	467	6	r.	r.	PROPN
ejde-761	467	7	;	;	PUNCT
ejde-761	467	8	hussain	hussain	PROPN
ejde-761	467	9	,	,	PUNCT
ejde-761	467	10	f.	f.	PROPN
ejde-761	467	11	;	;	PUNCT
ejde-761	467	12	ishtiaq	ishtiaq	PROPN
ejde-761	467	13	,	,	PUNCT
ejde-761	467	14	f.	f.	PROPN
ejde-761	467	15	;	;	PUNCT
ejde-761	467	16	et	et	PROPN
ejde-761	467	17	al	al	PROPN
ejde-761	467	18	.	.	PROPN
ejde-761	467	19	;	;	PUNCT
ejde-761	467	20	peristaltic	peristaltic	ADJ
ejde-761	467	21	transport	transport	NOUN
ejde-761	467	22	of	of	ADP
ejde-761	467	23	jeffrey	jeffrey	PROPN
ejde-761	467	24	fluid	fluid	NOUN
ejde-761	467	25	in	in	ADP
ejde-761	467	26	a	a	DET
ejde-761	467	27	rectangular	rectangular	ADJ
ejde-761	467	28	duct	duct	NOUN
ejde-761	467	29	through	through	ADP
ejde-761	467	30	a	a	DET
ejde-761	467	31	porous	porous	ADJ
ejde-761	467	32	medium	medium	NOUN
ejde-761	467	33	under	under	ADP
ejde-761	467	34	the	the	DET
ejde-761	467	35	effect	effect	NOUN
ejde-761	467	36	of	of	ADP
ejde-761	467	37	partial	partial	ADJ
ejde-761	467	38	slip	slip	NOUN
ejde-761	467	39	:	:	PUNCT
ejde-761	467	40	an	an	DET
ejde-761	467	41	application	application	NOUN
ejde-761	467	42	to	to	PART
ejde-761	467	43	upgrade	upgrade	VERB
ejde-761	467	44	industrial	industrial	ADJ
ejde-761	467	45	sieves	sieve	NOUN
ejde-761	467	46	/	/	SYM
ejde-761	467	47	filters	filter	NOUN
ejde-761	467	48	.	.	PUNCT
ejde-761	468	1	pramana	pramana	PROPN
ejde-761	468	2	j	j	PROPN
ejde-761	468	3	phys	phy	NOUN
ejde-761	468	4	93	93	NUM
ejde-761	468	5	(	(	PUNCT
ejde-761	468	6	2019	2019	NUM
ejde-761	468	7	)	)	PUNCT
ejde-761	468	8	,	,	PUNCT
ejde-761	468	9	34	34	NUM
ejde-761	468	10	.	.	PUNCT
ejde-761	468	11	doi	doi	PROPN
ejde-761	468	12	10.1007	10.1007	NUM
ejde-761	468	13	/	/	SYM
ejde-761	468	14	s12043	s12043	PROPN
ejde-761	468	15	-	-	PUNCT
ejde-761	468	16	019	019	NUM
ejde-761	468	17	-	-	PUNCT
ejde-761	468	18	1781	1781	NUM
ejde-761	468	19	-	-	SYM
ejde-761	468	20	8	8	NUM
ejde-761	469	1	[	[	X
ejde-761	469	2	21	21	NUM
ejde-761	469	3	]	]	X
ejde-761	469	4	enright	enright	PROPN
ejde-761	469	5	,	,	PUNCT
ejde-761	469	6	h.	h.	PROPN
ejde-761	469	7	;	;	PUNCT
ejde-761	469	8	muir	muir	PROPN
ejde-761	469	9	,	,	PUNCT
ejde-761	469	10	p.	p.	PROPN
ejde-761	469	11	h.	h.	PROPN
ejde-761	469	12	;	;	PUNCT
ejde-761	469	13	a	a	DET
ejde-761	469	14	runge	runge	NOUN
ejde-761	469	15	-	-	PUNCT
ejde-761	469	16	kutta	kutta	NOUN
ejde-761	469	17	type	type	NOUN
ejde-761	469	18	boundary	boundary	ADJ
ejde-761	469	19	value	value	NOUN
ejde-761	469	20	ode	ode	PROPN
ejde-761	469	21	solver	solver	NOUN
ejde-761	469	22	with	with	ADP
ejde-761	469	23	defect	defect	NOUN
ejde-761	469	24	control	control	NOUN
ejde-761	469	25	.	.	PUNCT
ejde-761	470	1	teh	teh	NOUN
ejde-761	470	2	.	.	PUNCT
ejde-761	471	1	rep	rep	PROPN
ejde-761	471	2	.	.	PROPN
ejde-761	471	3	267/93	267/93	NUM
ejde-761	471	4	,	,	PUNCT
ejde-761	471	5	university	university	NOUN
ejde-761	471	6	of	of	ADP
ejde-761	471	7	toronto	toronto	PROPN
ejde-761	471	8	,	,	PUNCT
ejde-761	471	9	dept	dept	PROPN
ejde-761	471	10	.	.	PROPN
ejde-761	471	11	of	of	ADP
ejde-761	471	12	computer	computer	NOUN
ejde-761	471	13	sciences	sciences	PROPN
ejde-761	471	14	.	.	PUNCT
ejde-761	472	1	toronto	toronto	PROPN
ejde-761	472	2	.	.	PUNCT
ejde-761	472	3	canada	canada	PROPN
ejde-761	472	4	,	,	PUNCT
ejde-761	472	5	1993	1993	NUM
ejde-761	472	6	.	.	PUNCT
ejde-761	473	1	[	[	X
ejde-761	473	2	22	22	NUM
ejde-761	473	3	]	]	X
ejde-761	473	4	fisher	fisher	PROPN
ejde-761	473	5	,	,	PUNCT
ejde-761	473	6	r.	r.	PROPN
ejde-761	473	7	a.	a.	PROPN
ejde-761	473	8	;	;	PUNCT
ejde-761	473	9	the	the	DET
ejde-761	473	10	advance	advance	NOUN
ejde-761	473	11	of	of	ADP
ejde-761	473	12	advantageous	advantageous	ADJ
ejde-761	473	13	genes	gene	NOUN
ejde-761	473	14	.	.	PUNCT
ejde-761	474	1	ann	ann	PROPN
ejde-761	474	2	.	.	PUNCT
ejde-761	474	3	eugenics	eugenic	NOUN
ejde-761	474	4	,	,	PUNCT
ejde-761	474	5	7	7	NUM
ejde-761	474	6	(	(	PUNCT
ejde-761	474	7	1937	1937	NUM
ejde-761	474	8	)	)	PUNCT
ejde-761	474	9	,	,	PUNCT
ejde-761	474	10	355–369	355–369	NUM
ejde-761	474	11	.	.	PUNCT
ejde-761	475	1	[	[	X
ejde-761	475	2	23	23	NUM
ejde-761	475	3	]	]	X
ejde-761	475	4	galaktionov	galaktionov	PROPN
ejde-761	475	5	,	,	PUNCT
ejde-761	475	6	v.	v.	PROPN
ejde-761	475	7	a.	a.	NOUN
ejde-761	475	8	;	;	PUNCT
ejde-761	475	9	three	three	NUM
ejde-761	475	10	types	type	NOUN
ejde-761	475	11	of	of	ADP
ejde-761	475	12	self	self	NOUN
ejde-761	475	13	-	-	PUNCT
ejde-761	475	14	similar	similar	ADJ
ejde-761	475	15	blow	blow	NOUN
ejde-761	475	16	-	-	PUNCT
ejde-761	475	17	up	up	NOUN
ejde-761	475	18	for	for	ADP
ejde-761	475	19	the	the	DET
ejde-761	475	20	fourth	fourth	ADJ
ejde-761	475	21	order	order	NOUN
ejde-761	475	22	p	p	NOUN
ejde-761	475	23	-	-	PUNCT
ejde-761	475	24	laplacian	laplacian	ADJ
ejde-761	475	25	equation	equation	NOUN
ejde-761	475	26	with	with	ADP
ejde-761	475	27	source	source	NOUN
ejde-761	475	28	.	.	PUNCT
ejde-761	476	1	jour	jour	AUX
ejde-761	476	2	.	.	PUNCT
ejde-761	476	3	comp	comp	PROPN
ejde-761	476	4	.	.	PUNCT
ejde-761	477	1	and	and	CCONJ
ejde-761	477	2	appl	appl	PROPN
ejde-761	477	3	.	.	PROPN
ejde-761	477	4	math	math	PROPN
ejde-761	477	5	.	.	PUNCT
ejde-761	478	1	,	,	PUNCT
ejde-761	478	2	223	223	NUM
ejde-761	478	3	(	(	PUNCT
ejde-761	478	4	2009	2009	NUM
ejde-761	478	5	)	)	PUNCT
ejde-761	478	6	,	,	PUNCT
ejde-761	478	7	326	326	NUM
ejde-761	478	8	-	-	SYM
ejde-761	478	9	355	355	NUM
ejde-761	478	10	.	.	PUNCT
ejde-761	479	1	[	[	X
ejde-761	479	2	24	24	NUM
ejde-761	479	3	]	]	PUNCT
ejde-761	479	4	galaktionov	galaktionov	PROPN
ejde-761	479	5	,	,	PUNCT
ejde-761	479	6	v.	v.	ADP
ejde-761	479	7	a.	a.	NOUN
ejde-761	479	8	;	;	PUNCT
ejde-761	479	9	on	on	ADP
ejde-761	479	10	a	a	DET
ejde-761	479	11	spectrum	spectrum	NOUN
ejde-761	479	12	of	of	ADP
ejde-761	479	13	blow	blow	NOUN
ejde-761	479	14	-	-	PUNCT
ejde-761	479	15	up	up	ADP
ejde-761	479	16	patterns	pattern	NOUN
ejde-761	479	17	for	for	ADP
ejde-761	479	18	a	a	DET
ejde-761	479	19	higher	high	ADJ
ejde-761	479	20	-	-	PUNCT
ejde-761	479	21	order	order	NOUN
ejde-761	479	22	semilinear	semilinear	PROPN
ejde-761	479	23	parabolic	parabolic	PROPN
ejde-761	479	24	equation	equation	NOUN
ejde-761	479	25	.	.	PUNCT
ejde-761	480	1	proceedings	proceeding	NOUN
ejde-761	480	2	of	of	ADP
ejde-761	480	3	the	the	DET
ejde-761	480	4	royal	royal	ADJ
ejde-761	480	5	society	society	NOUN
ejde-761	480	6	a	a	DET
ejde-761	480	7	:	:	PUNCT
ejde-761	480	8	mathematical	mathematical	ADJ
ejde-761	480	9	,	,	PUNCT
ejde-761	480	10	physical	physical	ADJ
ejde-761	480	11	and	and	CCONJ
ejde-761	480	12	engineering	engineering	NOUN
ejde-761	480	13	sciences	science	NOUN
ejde-761	480	14	.	.	PUNCT
ejde-761	481	1	2001	2001	NUM
ejde-761	481	2	.	.	PUNCT
ejde-761	482	1	[	[	X
ejde-761	482	2	25	25	NUM
ejde-761	482	3	]	]	X
ejde-761	482	4	galaktionov	galaktionov	PROPN
ejde-761	482	5	,	,	PUNCT
ejde-761	482	6	v.	v.	CCONJ
ejde-761	482	7	;	;	PUNCT
ejde-761	482	8	towards	towards	ADP
ejde-761	482	9	the	the	DET
ejde-761	482	10	kpp	kpp	ADJ
ejde-761	482	11	–	–	PUNCT
ejde-761	482	12	problem	problem	NOUN
ejde-761	482	13	and	and	CCONJ
ejde-761	482	14	log	log	NOUN
ejde-761	482	15	-	-	PUNCT
ejde-761	482	16	front	front	NOUN
ejde-761	482	17	shift	shift	NOUN
ejde-761	482	18	for	for	ADP
ejde-761	482	19	higher	high	ADJ
ejde-761	482	20	-	-	PUNCT
ejde-761	482	21	order	order	NOUN
ejde-761	482	22	nonlinear	nonlinear	ADJ
ejde-761	482	23	pdes	pde	NOUN
ejde-761	482	24	i.	i.	PROPN
ejde-761	482	25	bi	bi	PROPN
ejde-761	482	26	-	-	ADJ
ejde-761	482	27	harmonic	harmonic	ADJ
ejde-761	482	28	and	and	CCONJ
ejde-761	482	29	other	other	ADJ
ejde-761	482	30	parabolic	parabolic	ADJ
ejde-761	482	31	equations	equation	NOUN
ejde-761	482	32	.	.	PUNCT
ejde-761	483	1	cornwell	cornwell	PROPN
ejde-761	483	2	university	university	PROPN
ejde-761	483	3	arxiv:1210.3513	arxiv:1210.3513	PROPN
ejde-761	483	4	,	,	PUNCT
ejde-761	483	5	2012	2012	NUM
ejde-761	483	6	.	.	PUNCT
ejde-761	484	1	[	[	X
ejde-761	484	2	26	26	NUM
ejde-761	484	3	]	]	X
ejde-761	484	4	galaktionov	galaktionov	PROPN
ejde-761	484	5	,	,	PUNCT
ejde-761	484	6	v.	v.	PROPN
ejde-761	484	7	;	;	PUNCT
ejde-761	484	8	shishkov	shishkov	PROPN
ejde-761	484	9	,	,	PUNCT
ejde-761	484	10	a.	a.	NOUN
ejde-761	484	11	;	;	PUNCT
ejde-761	484	12	saint	saint	PROPN
ejde-761	484	13	-	-	PUNCT
ejde-761	484	14	venant	venant	NOUN
ejde-761	484	15	’s	’s	PART
ejde-761	484	16	principle	principle	NOUN
ejde-761	484	17	in	in	ADP
ejde-761	484	18	blow	blow	NOUN
ejde-761	484	19	-	-	PUNCT
ejde-761	484	20	up	up	NOUN
ejde-761	484	21	for	for	ADP
ejde-761	484	22	higher	high	ADJ
ejde-761	484	23	-	-	PUNCT
ejde-761	484	24	order	order	NOUN
ejde-761	484	25	quasilinear	quasilinear	NOUN
ejde-761	484	26	parabolic	parabolic	PROPN
ejde-761	484	27	equations	equation	NOUN
ejde-761	484	28	.	.	PUNCT
ejde-761	485	1	proceedings	proceeding	NOUN
ejde-761	485	2	of	of	ADP
ejde-761	485	3	the	the	DET
ejde-761	485	4	royal	royal	ADJ
ejde-761	485	5	society	society	NOUN
ejde-761	485	6	of	of	ADP
ejde-761	485	7	edinburgh	edinburgh	PROPN
ejde-761	485	8	:	:	PUNCT
ejde-761	485	9	section	section	VERB
ejde-761	485	10	a	a	DET
ejde-761	485	11	mathematics	mathematic	NOUN
ejde-761	485	12	,	,	PUNCT
ejde-761	485	13	133(5	133(5	NUM
ejde-761	485	14	)	)	PUNCT
ejde-761	485	15	(	(	PUNCT
ejde-761	485	16	2003	2003	NUM
ejde-761	485	17	)	)	PUNCT
ejde-761	485	18	,	,	PUNCT
ejde-761	485	19	1075	1075	NUM
ejde-761	485	20	-	-	SYM
ejde-761	485	21	1119	1119	NUM
ejde-761	485	22	.	.	PUNCT
ejde-761	486	1	doi	doi	PROPN
ejde-761	486	2	10.1017	10.1017	NUM
ejde-761	486	3	/	/	SYM
ejde-761	486	4	s0308210500002821	s0308210500002821	PROPN
ejde-761	486	5	.	.	PUNCT
ejde-761	487	1	[	[	X
ejde-761	487	2	27	27	NUM
ejde-761	487	3	]	]	X
ejde-761	487	4	goldshtein	goldshtein	NOUN
ejde-761	487	5	,	,	PUNCT
ejde-761	487	6	v.	v.	ADV
ejde-761	487	7	;	;	PUNCT
ejde-761	487	8	ukhlov	ukhlov	ADJ
ejde-761	487	9	,	,	PUNCT
ejde-761	487	10	a.	a.	NOUN
ejde-761	487	11	;	;	PUNCT
ejde-761	487	12	weighted	weight	VERB
ejde-761	487	13	sobolev	sobolev	NOUN
ejde-761	487	14	spaces	space	NOUN
ejde-761	487	15	and	and	CCONJ
ejde-761	487	16	embeddings	embedding	NOUN
ejde-761	487	17	theorems	theorem	NOUN
ejde-761	487	18	.	.	PUNCT
ejde-761	488	1	transactions	transaction	NOUN
ejde-761	488	2	of	of	ADP
ejde-761	488	3	the	the	DET
ejde-761	488	4	american	american	PROPN
ejde-761	488	5	mathematical	mathematical	PROPN
ejde-761	488	6	society	society	NOUN
ejde-761	488	7	.	.	PUNCT
ejde-761	489	1	361	361	NUM
ejde-761	489	2	(	(	PUNCT
ejde-761	489	3	2009	2009	NUM
ejde-761	489	4	)	)	PUNCT
ejde-761	489	5	,	,	PUNCT
ejde-761	489	6	3829	3829	NUM
ejde-761	489	7	-	-	SYM
ejde-761	489	8	3850	3850	NUM
ejde-761	489	9	.	.	PUNCT
ejde-761	490	1	[	[	X
ejde-761	490	2	28	28	NUM
ejde-761	490	3	]	]	X
ejde-761	490	4	hongjun	hongjun	NOUN
ejde-761	490	5	,	,	PUNCT
ejde-761	490	6	g.	g.	PROPN
ejde-761	490	7	;	;	PUNCT
ejde-761	490	8	changchun	changchun	PROPN
ejde-761	490	9	,	,	PUNCT
ejde-761	490	10	l.	l.	PROPN
ejde-761	490	11	;	;	PUNCT
ejde-761	490	12	instabilities	instability	NOUN
ejde-761	490	13	of	of	ADP
ejde-761	490	14	traveling	travel	VERB
ejde-761	490	15	waves	wave	NOUN
ejde-761	490	16	of	of	ADP
ejde-761	490	17	the	the	DET
ejde-761	490	18	convective	convective	ADJ
ejde-761	490	19	-	-	PUNCT
ejde-761	490	20	diffusive	diffusive	ADJ
ejde-761	490	21	cahnhilliard	cahnhilliard	NOUN
ejde-761	490	22	equation	equation	NOUN
ejde-761	490	23	.	.	PUNCT
ejde-761	491	1	chaos	chaos	NOUN
ejde-761	491	2	,	,	PUNCT
ejde-761	491	3	solitons	soliton	NOUN
ejde-761	491	4	and	and	CCONJ
ejde-761	491	5	fractals	fractal	NOUN
ejde-761	491	6	,	,	PUNCT
ejde-761	491	7	20	20	NUM
ejde-761	491	8	(	(	PUNCT
ejde-761	491	9	2004	2004	NUM
ejde-761	491	10	)	)	PUNCT
ejde-761	491	11	,	,	PUNCT
ejde-761	491	12	253	253	NUM
ejde-761	491	13	-	-	SYM
ejde-761	491	14	258	258	NUM
ejde-761	491	15	.	.	PUNCT
ejde-761	492	1	[	[	X
ejde-761	492	2	29	29	NUM
ejde-761	492	3	]	]	X
ejde-761	492	4	kamin	kamin	PROPN
ejde-761	492	5	,	,	PUNCT
ejde-761	492	6	s.	s.	PROPN
ejde-761	492	7	;	;	PUNCT
ejde-761	492	8	vázquez	vázquez	PROPN
ejde-761	492	9	,	,	PUNCT
ejde-761	492	10	j.	j.	PROPN
ejde-761	492	11	l.	l.	PROPN
ejde-761	492	12	;	;	PUNCT
ejde-761	492	13	fundamental	fundamental	ADJ
ejde-761	492	14	solutions	solution	NOUN
ejde-761	492	15	and	and	CCONJ
ejde-761	492	16	asymptotic	asymptotic	ADJ
ejde-761	492	17	behaviour	behaviour	NOUN
ejde-761	492	18	for	for	ADP
ejde-761	492	19	the	the	DET
ejde-761	492	20	plaplacian	plaplacian	ADJ
ejde-761	492	21	equation	equation	NOUN
ejde-761	492	22	.	.	PUNCT
ejde-761	493	1	revista	revista	PROPN
ejde-761	493	2	matemática	matemática	PROPN
ejde-761	493	3	iberoamericana	iberoamericana	PROPN
ejde-761	493	4	.	.	PUNCT
ejde-761	494	1	vol	vol	NOUN
ejde-761	494	2	4	4	NUM
ejde-761	494	3	(	(	PUNCT
ejde-761	494	4	1988	1988	NUM
ejde-761	494	5	)	)	PUNCT
ejde-761	494	6	,	,	PUNCT
ejde-761	494	7	2	2	X
ejde-761	494	8	.	.	PUNCT
ejde-761	495	1	[	[	X
ejde-761	495	2	30	30	NUM
ejde-761	495	3	]	]	X
ejde-761	495	4	keller	keller	PROPN
ejde-761	495	5	,	,	PUNCT
ejde-761	495	6	e.	e.	PROPN
ejde-761	495	7	f.	f.	PROPN
ejde-761	495	8	;	;	PUNCT
ejde-761	495	9	segel	segel	PROPN
ejde-761	495	10	,	,	PUNCT
ejde-761	495	11	l.	l.	PROPN
ejde-761	495	12	a.	a.	PROPN
ejde-761	495	13	;	;	PUNCT
ejde-761	495	14	traveling	travel	VERB
ejde-761	495	15	bands	band	NOUN
ejde-761	495	16	of	of	ADP
ejde-761	495	17	chemotactic	chemotactic	ADJ
ejde-761	495	18	bacteria	bacteria	NOUN
ejde-761	495	19	:	:	PUNCT
ejde-761	495	20	a	a	DET
ejde-761	495	21	theoretical	theoretical	ADJ
ejde-761	495	22	analysis	analysis	NOUN
ejde-761	495	23	.	.	PUNCT
ejde-761	496	1	j.	j.	PROPN
ejde-761	496	2	theoret	theoret	PROPN
ejde-761	496	3	.	.	PUNCT
ejde-761	497	1	biol	biol	PROPN
ejde-761	497	2	.	.	PUNCT
ejde-761	498	1	30	30	NUM
ejde-761	498	2	(	(	PUNCT
ejde-761	498	3	1971	1971	NUM
ejde-761	498	4	)	)	PUNCT
ejde-761	498	5	,	,	PUNCT
ejde-761	498	6	235	235	NUM
ejde-761	498	7	-	-	SYM
ejde-761	498	8	248	248	NUM
ejde-761	498	9	.	.	PUNCT
ejde-761	499	1	22	22	NUM
ejde-761	499	2	j.	j.	PROPN
ejde-761	499	3	l.	l.	PROPN
ejde-761	499	4	díaz	díaz	PROPN
ejde-761	499	5	palencia	palencia	PROPN
ejde-761	499	6	ejde-2024/68	ejde-2024/68	PROPN
ejde-761	499	7	[	[	X
ejde-761	499	8	31	31	NUM
ejde-761	499	9	]	]	PUNCT
ejde-761	499	10	kesavan	kesavan	NOUN
ejde-761	499	11	,	,	PUNCT
ejde-761	499	12	s.	s.	PROPN
ejde-761	499	13	;	;	PUNCT
ejde-761	499	14	topics	topic	NOUN
ejde-761	499	15	in	in	ADP
ejde-761	499	16	functional	functional	ADJ
ejde-761	499	17	analysis	analysis	NOUN
ejde-761	499	18	and	and	CCONJ
ejde-761	499	19	applications	application	NOUN
ejde-761	499	20	,	,	PUNCT
ejde-761	499	21	new	new	ADJ
ejde-761	499	22	age	age	NOUN
ejde-761	499	23	international	international	NOUN
ejde-761	499	24	(	(	PUNCT
ejde-761	499	25	formerly	formerly	ADV
ejde-761	499	26	wiley	wiley	NOUN
ejde-761	499	27	-	-	PUNCT
ejde-761	499	28	eastern	eastern	ADJ
ejde-761	499	29	)	)	PUNCT
ejde-761	499	30	.	.	PUNCT
ejde-761	500	1	1989	1989	NUM
ejde-761	500	2	.	.	PUNCT
ejde-761	501	1	[	[	X
ejde-761	501	2	32	32	NUM
ejde-761	501	3	]	]	SYM
ejde-761	501	4	kolmogorov	kolmogorov	PROPN
ejde-761	501	5	,	,	PUNCT
ejde-761	501	6	a	a	DET
ejde-761	501	7	.n	.n	NOUN
ejde-761	501	8	.	.	PUNCT
ejde-761	501	9	;	;	PUNCT
ejde-761	501	10	petrovskii	petrovskii	PROPN
ejde-761	501	11	,	,	PUNCT
ejde-761	501	12	i.	i.	PROPN
ejde-761	501	13	g.	g.	PROPN
ejde-761	501	14	;	;	PUNCT
ejde-761	501	15	piskunov	piskunov	PROPN
ejde-761	501	16	,	,	PUNCT
ejde-761	501	17	n.	n.	PROPN
ejde-761	501	18	s.	s.	PROPN
ejde-761	501	19	;	;	PUNCT
ejde-761	501	20	study	study	NOUN
ejde-761	501	21	of	of	ADP
ejde-761	501	22	the	the	DET
ejde-761	501	23	diffusion	diffusion	NOUN
ejde-761	501	24	equation	equation	NOUN
ejde-761	501	25	with	with	ADP
ejde-761	501	26	growth	growth	NOUN
ejde-761	501	27	of	of	ADP
ejde-761	501	28	the	the	DET
ejde-761	501	29	quantity	quantity	NOUN
ejde-761	501	30	of	of	ADP
ejde-761	501	31	matter	matter	NOUN
ejde-761	501	32	and	and	CCONJ
ejde-761	501	33	its	its	PRON
ejde-761	501	34	application	application	NOUN
ejde-761	501	35	to	to	ADP
ejde-761	501	36	a	a	DET
ejde-761	501	37	biological	biological	ADJ
ejde-761	501	38	problem	problem	NOUN
ejde-761	501	39	.	.	PUNCT
ejde-761	502	1	byull	byull	NOUN
ejde-761	502	2	.	.	PUNCT
ejde-761	503	1	moskov	moskov	PROPN
ejde-761	503	2	.	.	PUNCT
ejde-761	504	1	gos	gos	PROPN
ejde-761	504	2	.	.	PUNCT
ejde-761	505	1	univ	univ	PROPN
ejde-761	505	2	.	.	PROPN
ejde-761	505	3	,	,	PUNCT
ejde-761	505	4	sect	sect	NOUN
ejde-761	505	5	.	.	PUNCT
ejde-761	506	1	a	a	DET
ejde-761	506	2	,	,	PUNCT
ejde-761	506	3	1	1	NUM
ejde-761	506	4	(	(	PUNCT
ejde-761	506	5	1937	1937	NUM
ejde-761	506	6	)	)	PUNCT
ejde-761	506	7	.	.	PUNCT
ejde-761	507	1	[	[	X
ejde-761	507	2	33	33	NUM
ejde-761	507	3	]	]	X
ejde-761	507	4	ladyzhenskaya	ladyzhenskaya	PROPN
ejde-761	507	5	,	,	PUNCT
ejde-761	507	6	o.	o.	PROPN
ejde-761	507	7	;	;	PUNCT
ejde-761	507	8	some	some	PRON
ejde-761	507	9	results	result	VERB
ejde-761	507	10	on	on	ADP
ejde-761	507	11	modifications	modification	NOUN
ejde-761	507	12	of	of	ADP
ejde-761	507	13	three	three	NUM
ejde-761	507	14	-	-	PUNCT
ejde-761	507	15	dimensional	dimensional	ADJ
ejde-761	507	16	navier	navier	NOUN
ejde-761	507	17	-	-	PUNCT
ejde-761	507	18	stokes	stoke	NOUN
ejde-761	507	19	equations	equation	NOUN
ejde-761	507	20	.	.	PUNCT
ejde-761	508	1	nonlinear	nonlinear	ADJ
ejde-761	508	2	analysis	analysis	NOUN
ejde-761	508	3	and	and	CCONJ
ejde-761	508	4	continuum	continuum	ADJ
ejde-761	508	5	mechanics	mechanic	NOUN
ejde-761	508	6	.	.	PUNCT
ejde-761	509	1	(	(	PUNCT
ejde-761	509	2	1998	1998	NUM
ejde-761	509	3	)	)	PUNCT
ejde-761	509	4	pp	pp	ADP
ejde-761	509	5	.	.	PUNCT
ejde-761	510	1	73–84	73–84	NOUN
ejde-761	510	2	.	.	PUNCT
ejde-761	511	1	[	[	X
ejde-761	511	2	34	34	NUM
ejde-761	511	3	]	]	SYM
ejde-761	511	4	li	li	PROPN
ejde-761	511	5	,	,	PUNCT
ejde-761	511	6	zhenbang	zhenbang	PROPN
ejde-761	511	7	;	;	PUNCT
ejde-761	511	8	liu	liu	PROPN
ejde-761	511	9	,	,	PUNCT
ejde-761	511	10	changchun	changchun	PROPN
ejde-761	511	11	;	;	PUNCT
ejde-761	511	12	on	on	ADP
ejde-761	511	13	the	the	DET
ejde-761	511	14	nonlinear	nonlinear	ADJ
ejde-761	511	15	instability	instability	NOUN
ejde-761	511	16	of	of	ADP
ejde-761	511	17	traveling	travel	VERB
ejde-761	511	18	waves	wave	NOUN
ejde-761	511	19	for	for	ADP
ejde-761	511	20	a	a	DET
ejde-761	511	21	sixthorder	sixthorder	NOUN
ejde-761	511	22	parabolic	parabolic	NOUN
ejde-761	511	23	equation	equation	NOUN
ejde-761	511	24	.	.	PUNCT
ejde-761	512	1	abstr	abstr	PROPN
ejde-761	512	2	.	.	PUNCT
ejde-761	512	3	appl	appl	PROPN
ejde-761	512	4	.	.	PUNCT
ejde-761	513	1	anal	anal	PROPN
ejde-761	513	2	.	.	PUNCT
ejde-761	514	1	article	article	NOUN
ejde-761	514	2	i	i	PROPN
ejde-761	514	3	d	d	PROPN
ejde-761	514	4	739156	739156	NUM
ejde-761	514	5	(	(	PUNCT
ejde-761	514	6	2012	2012	NUM
ejde-761	514	7	)	)	PUNCT
ejde-761	514	8	,	,	PUNCT
ejde-761	514	9	17	17	NUM
ejde-761	514	10	pages	page	NOUN
ejde-761	514	11	.	.	PUNCT
ejde-761	515	1	doi	doi	PROPN
ejde-761	515	2	10.1155/2012/739156	10.1155/2012/739156	NUM
ejde-761	516	1	[	[	X
ejde-761	516	2	35	35	NUM
ejde-761	516	3	]	]	X
ejde-761	516	4	montaru	montaru	NOUN
ejde-761	516	5	,	,	PUNCT
ejde-761	516	6	a.	a.	NOUN
ejde-761	516	7	;	;	PUNCT
ejde-761	516	8	wellposedness	wellposedness	NOUN
ejde-761	516	9	and	and	CCONJ
ejde-761	516	10	regularity	regularity	NOUN
ejde-761	516	11	for	for	ADP
ejde-761	516	12	a	a	DET
ejde-761	516	13	degenerate	degenerate	ADJ
ejde-761	516	14	parabolic	parabolic	NOUN
ejde-761	516	15	equation	equation	NOUN
ejde-761	516	16	arising	arise	VERB
ejde-761	516	17	in	in	ADP
ejde-761	516	18	a	a	DET
ejde-761	516	19	model	model	NOUN
ejde-761	516	20	of	of	ADP
ejde-761	516	21	chemotaxis	chemotaxis	ADJ
ejde-761	516	22	with	with	ADP
ejde-761	516	23	nonlinear	nonlinear	ADJ
ejde-761	516	24	sensitivity	sensitivity	NOUN
ejde-761	516	25	.	.	PUNCT
ejde-761	517	1	discrete	discrete	ADJ
ejde-761	517	2	and	and	CCONJ
ejde-761	517	3	continuous	continuous	ADJ
ejde-761	517	4	dynamical	dynamical	ADJ
ejde-761	517	5	systems	system	NOUN
ejde-761	517	6	b	b	PROPN
ejde-761	517	7	,	,	PUNCT
ejde-761	517	8	19,1	19,1	PROPN
ejde-761	517	9	(	(	PUNCT
ejde-761	517	10	2014	2014	NUM
ejde-761	517	11	)	)	PUNCT
ejde-761	517	12	,	,	PUNCT
ejde-761	517	13	231	231	NUM
ejde-761	517	14	-	-	SYM
ejde-761	517	15	256	256	NUM
ejde-761	517	16	.	.	PUNCT
ejde-761	518	1	[	[	X
ejde-761	518	2	36	36	NUM
ejde-761	518	3	]	]	X
ejde-761	518	4	niemela	niemela	NOUN
ejde-761	518	5	,	,	PUNCT
ejde-761	518	6	j.	j.	PROPN
ejde-761	518	7	j.	j.	PROPN
ejde-761	518	8	;	;	PUNCT
ejde-761	518	9	ahlers	ahler	NOUN
ejde-761	518	10	,	,	PUNCT
ejde-761	518	11	g.	g.	PROPN
ejde-761	518	12	;	;	PUNCT
ejde-761	518	13	cannell	cannell	PROPN
ejde-761	518	14	,	,	PUNCT
ejde-761	518	15	d.	d.	PROPN
ejde-761	518	16	s.	s.	PROPN
ejde-761	518	17	;	;	PUNCT
ejde-761	518	18	localized	localized	ADJ
ejde-761	518	19	traveling	travel	VERB
ejde-761	518	20	-	-	PUNCT
ejde-761	518	21	wave	wave	NOUN
ejde-761	518	22	states	state	NOUN
ejde-761	518	23	in	in	ADP
ejde-761	518	24	binary	binary	ADJ
ejde-761	518	25	-	-	PUNCT
ejde-761	518	26	fluid	fluid	NOUN
ejde-761	518	27	convection	convection	NOUN
ejde-761	518	28	.	.	PUNCT
ejde-761	519	1	phys	phy	NOUN
ejde-761	519	2	.	.	PUNCT
ejde-761	520	1	rev	rev	PROPN
ejde-761	520	2	.	.	PROPN
ejde-761	520	3	lett	lett	PROPN
ejde-761	520	4	.	.	PUNCT
ejde-761	521	1	64	64	NUM
ejde-761	521	2	(	(	PUNCT
ejde-761	521	3	1990	1990	NUM
ejde-761	521	4	)	)	PUNCT
ejde-761	521	5	,	,	PUNCT
ejde-761	521	6	1365–1368	1365–1368	NUM
ejde-761	521	7	.	.	PUNCT
ejde-761	522	1	doi	doi	PROPN
ejde-761	522	2	10.1103	10.1103	NUM
ejde-761	522	3	/	/	SYM
ejde-761	522	4	physrevlett.64.1365	physrevlett.64.1365	NOUN
ejde-761	522	5	.	.	PUNCT
ejde-761	523	1	[	[	X
ejde-761	523	2	37	37	NUM
ejde-761	523	3	]	]	SYM
ejde-761	523	4	okubo	okubo	NOUN
ejde-761	523	5	,	,	PUNCT
ejde-761	523	6	a.	a.	NOUN
ejde-761	523	7	;	;	PUNCT
ejde-761	523	8	levin	levin	PROPN
ejde-761	523	9	,	,	PUNCT
ejde-761	523	10	s.	s.	PROPN
ejde-761	523	11	a.	a.	PROPN
ejde-761	523	12	;	;	PUNCT
ejde-761	523	13	the	the	DET
ejde-761	523	14	basics	basic	NOUN
ejde-761	523	15	of	of	ADP
ejde-761	523	16	diffusion	diffusion	NOUN
ejde-761	523	17	.	.	PUNCT
ejde-761	524	1	in	in	ADP
ejde-761	524	2	:	:	PUNCT
ejde-761	524	3	diffusion	diffusion	NOUN
ejde-761	524	4	and	and	CCONJ
ejde-761	524	5	ecological	ecological	ADJ
ejde-761	524	6	problems	problem	NOUN
ejde-761	524	7	:	:	PUNCT
ejde-761	524	8	modern	modern	ADJ
ejde-761	524	9	perspectives	perspective	NOUN
ejde-761	524	10	.	.	PUNCT
ejde-761	525	1	interdisciplinary	interdisciplinary	ADJ
ejde-761	525	2	applied	apply	VERB
ejde-761	525	3	mathematics	mathematic	NOUN
ejde-761	525	4	,	,	PUNCT
ejde-761	525	5	vol	vol	NOUN
ejde-761	525	6	14	14	NUM
ejde-761	525	7	.	.	PUNCT
ejde-761	525	8	springer	springer	NOUN
ejde-761	525	9	,	,	PUNCT
ejde-761	525	10	new	new	PROPN
ejde-761	525	11	york	york	PROPN
ejde-761	525	12	,	,	PUNCT
ejde-761	525	13	ny	ny	PROPN
ejde-761	525	14	,	,	PUNCT
ejde-761	525	15	2001	2001	NUM
ejde-761	525	16	.	.	PUNCT
ejde-761	526	1	doi	doi	PROPN
ejde-761	526	2	10.1007/978	10.1007/978	NUM
ejde-761	526	3	-	-	SYM
ejde-761	526	4	1	1	NUM
ejde-761	526	5	-	-	PUNCT
ejde-761	526	6	4757	4757	NUM
ejde-761	526	7	-	-	PUNCT
ejde-761	526	8	4978	4978	NUM
ejde-761	526	9	-	-	SYM
ejde-761	526	10	6	6	NUM
ejde-761	526	11	.	.	PUNCT
ejde-761	527	1	[	[	X
ejde-761	527	2	38	38	NUM
ejde-761	527	3	]	]	PUNCT
ejde-761	527	4	palencia	palencia	PROPN
ejde-761	527	5	,	,	PUNCT
ejde-761	527	6	j.	j.	PROPN
ejde-761	527	7	l.	l.	PROPN
ejde-761	527	8	d.	d.	PROPN
ejde-761	527	9	;	;	PUNCT
ejde-761	527	10	analysis	analysis	NOUN
ejde-761	527	11	of	of	ADP
ejde-761	527	12	selfsimilar	selfsimilar	ADJ
ejde-761	527	13	solutions	solution	NOUN
ejde-761	527	14	and	and	CCONJ
ejde-761	527	15	a	a	DET
ejde-761	527	16	comparison	comparison	NOUN
ejde-761	527	17	principle	principle	NOUN
ejde-761	527	18	for	for	ADP
ejde-761	527	19	an	an	DET
ejde-761	527	20	heterogeneous	heterogeneous	ADJ
ejde-761	527	21	diffusion	diffusion	NOUN
ejde-761	527	22	cooperative	cooperative	ADJ
ejde-761	527	23	system	system	NOUN
ejde-761	527	24	with	with	ADP
ejde-761	527	25	advection	advection	NOUN
ejde-761	527	26	and	and	CCONJ
ejde-761	527	27	non	non	ADJ
ejde-761	527	28	-	-	ADJ
ejde-761	527	29	linear	linear	ADJ
ejde-761	527	30	reaction	reaction	NOUN
ejde-761	527	31	.	.	PUNCT
ejde-761	528	1	comp	comp	PROPN
ejde-761	528	2	.	.	PUNCT
ejde-761	529	1	appl	appl	PROPN
ejde-761	529	2	.	.	PROPN
ejde-761	529	3	math	math	NOUN
ejde-761	529	4	.	.	PUNCT
ejde-761	530	1	40	40	NUM
ejde-761	530	2	(	(	PUNCT
ejde-761	530	3	2021	2021	NUM
ejde-761	530	4	)	)	PUNCT
ejde-761	530	5	,	,	PUNCT
ejde-761	530	6	302	302	NUM
ejde-761	530	7	.	.	PUNCT
ejde-761	531	1	doi	doi	NOUN
ejde-761	531	2	10.1007	10.1007	NUM
ejde-761	531	3	/	/	SYM
ejde-761	531	4	s40314	s40314	NOUN
ejde-761	531	5	-	-	PUNCT
ejde-761	531	6	021	021	NUM
ejde-761	531	7	-	-	PUNCT
ejde-761	531	8	01689	01689	NUM
ejde-761	531	9	-	-	PUNCT
ejde-761	531	10	y	y	NOUN
ejde-761	532	1	[	[	X
ejde-761	532	2	39	39	NUM
ejde-761	532	3	]	]	PUNCT
ejde-761	532	4	peletier	peletier	NOUN
ejde-761	532	5	,	,	PUNCT
ejde-761	532	6	l.	l.	PROPN
ejde-761	532	7	a.	a.	PROPN
ejde-761	532	8	;	;	PUNCT
ejde-761	532	9	troy	troy	PROPN
ejde-761	532	10	,	,	PUNCT
ejde-761	532	11	w.	w.	PROPN
ejde-761	532	12	c.	c.	PROPN
ejde-761	532	13	;	;	PUNCT
ejde-761	532	14	spatial	spatial	ADJ
ejde-761	532	15	patterns	pattern	NOUN
ejde-761	532	16	.	.	PUNCT
ejde-761	533	1	higher	high	ADJ
ejde-761	533	2	order	order	NOUN
ejde-761	533	3	models	model	NOUN
ejde-761	533	4	in	in	ADP
ejde-761	533	5	physics	physics	NOUN
ejde-761	533	6	and	and	CCONJ
ejde-761	533	7	mechanics	mechanic	NOUN
ejde-761	533	8	.	.	PUNCT
ejde-761	534	1	progress	progress	NOUN
ejde-761	534	2	in	in	ADP
ejde-761	534	3	non	non	ADJ
ejde-761	534	4	linear	linear	PROPN
ejde-761	534	5	differential	differential	ADJ
ejde-761	534	6	equations	equation	NOUN
ejde-761	534	7	and	and	CCONJ
ejde-761	534	8	their	their	PRON
ejde-761	534	9	applications	application	NOUN
ejde-761	534	10	.	.	PUNCT
ejde-761	535	1	volume	volume	NOUN
ejde-761	535	2	45	45	NUM
ejde-761	535	3	.	.	PUNCT
ejde-761	536	1	universit’e	universit’e	PROPN
ejde-761	536	2	pierre	pierre	PROPN
ejde-761	536	3	et	et	PROPN
ejde-761	536	4	marie	marie	PROPN
ejde-761	536	5	curie	curie	PROPN
ejde-761	536	6	,	,	PUNCT
ejde-761	536	7	2001	2001	NUM
ejde-761	536	8	.	.	PUNCT
ejde-761	537	1	[	[	X
ejde-761	537	2	40	40	NUM
ejde-761	537	3	]	]	X
ejde-761	537	4	rauprich	rauprich	NOUN
ejde-761	537	5	,	,	PUNCT
ejde-761	537	6	o.	o.	PROPN
ejde-761	537	7	;	;	PUNCT
ejde-761	537	8	matsushita	matsushita	PROPN
ejde-761	537	9	,	,	PUNCT
ejde-761	537	10	m.	m.	NOUN
ejde-761	537	11	;	;	PUNCT
ejde-761	537	12	weijer	weijer	NOUN
ejde-761	537	13	,	,	PUNCT
ejde-761	537	14	c.	c.	PROPN
ejde-761	537	15	j.	j.	PROPN
ejde-761	537	16	;	;	PUNCT
ejde-761	537	17	siegert	siegert	PROPN
ejde-761	537	18	,	,	PUNCT
ejde-761	537	19	f.	f.	PROPN
ejde-761	537	20	;	;	PUNCT
ejde-761	537	21	esipov	esipov	PROPN
ejde-761	537	22	,	,	PUNCT
ejde-761	537	23	s.	s.	PROPN
ejde-761	537	24	e.	e.	PROPN
ejde-761	537	25	;	;	PUNCT
ejde-761	537	26	shapiro	shapiro	PROPN
ejde-761	537	27	,	,	PUNCT
ejde-761	537	28	j.	j.	PROPN
ejde-761	537	29	a.	a.	PROPN
ejde-761	537	30	;	;	PUNCT
ejde-761	537	31	periodic	periodic	ADJ
ejde-761	537	32	phenomena	phenomenon	NOUN
ejde-761	537	33	in	in	ADP
ejde-761	537	34	proteus	proteus	NOUN
ejde-761	537	35	mirabilis	mirabilis	PROPN
ejde-761	537	36	swarm	swarm	NOUN
ejde-761	537	37	colony	colony	NOUN
ejde-761	537	38	development	development	NOUN
ejde-761	537	39	.	.	PUNCT
ejde-761	538	1	j.	j.	PROPN
ejde-761	538	2	bacteriol	bacteriol	PROPN
ejde-761	538	3	.	.	PUNCT
ejde-761	539	1	178	178	NUM
ejde-761	539	2	(	(	PUNCT
ejde-761	539	3	1996	1996	NUM
ejde-761	539	4	)	)	PUNCT
ejde-761	539	5	,	,	PUNCT
ejde-761	539	6	6525–6538	6525–6538	NUM
ejde-761	539	7	.	.	PUNCT
ejde-761	540	1	doi	doi	NOUN
ejde-761	540	2	10.1128	10.1128	NUM
ejde-761	540	3	/	/	SYM
ejde-761	540	4	jb.178.22.6525	jb.178.22.6525	NOUN
ejde-761	540	5	-	-	PUNCT
ejde-761	540	6	6538.1996	6538.1996	NOUN
ejde-761	540	7	[	[	X
ejde-761	540	8	41	41	NUM
ejde-761	540	9	]	]	X
ejde-761	540	10	rottschäfer	rottschäfer	NOUN
ejde-761	540	11	,	,	PUNCT
ejde-761	540	12	v.	v.	ADV
ejde-761	540	13	;	;	PUNCT
ejde-761	540	14	doelman	doelman	NOUN
ejde-761	540	15	,	,	PUNCT
ejde-761	540	16	a.	a.	NOUN
ejde-761	540	17	;	;	PUNCT
ejde-761	540	18	on	on	ADP
ejde-761	540	19	the	the	DET
ejde-761	540	20	transition	transition	NOUN
ejde-761	540	21	from	from	ADP
ejde-761	540	22	the	the	DET
ejde-761	540	23	ginzburg	ginzburg	NOUN
ejde-761	540	24	-	-	PUNCT
ejde-761	540	25	landau	landau	NOUN
ejde-761	540	26	equation	equation	NOUN
ejde-761	540	27	to	to	ADP
ejde-761	540	28	the	the	DET
ejde-761	540	29	extended	extended	ADJ
ejde-761	540	30	fisher	fisher	PROPN
ejde-761	540	31	-	-	PUNCT
ejde-761	540	32	kolmogorov	kolmogorov	PROPN
ejde-761	540	33	equation	equation	NOUN
ejde-761	540	34	.	.	PUNCT
ejde-761	541	1	physica	physica	PROPN
ejde-761	541	2	d.	d.	PROPN
ejde-761	541	3	118	118	NUM
ejde-761	541	4	(	(	PUNCT
ejde-761	541	5	1998	1998	NUM
ejde-761	541	6	)	)	PUNCT
ejde-761	541	7	,	,	PUNCT
ejde-761	541	8	261	261	NUM
ejde-761	541	9	-	-	SYM
ejde-761	541	10	292	292	NUM
ejde-761	541	11	.	.	PUNCT
ejde-761	542	1	[	[	X
ejde-761	542	2	42	42	NUM
ejde-761	542	3	]	]	SYM
ejde-761	542	4	shishkov	shishkov	PROPN
ejde-761	542	5	,	,	PUNCT
ejde-761	542	6	a.	a.	PROPN
ejde-761	542	7	e.	e.	PROPN
ejde-761	542	8	;	;	PUNCT
ejde-761	542	9	dead	dead	ADJ
ejde-761	542	10	cores	core	NOUN
ejde-761	542	11	and	and	CCONJ
ejde-761	542	12	instantaneous	instantaneous	ADJ
ejde-761	542	13	compactification	compactification	NOUN
ejde-761	542	14	of	of	ADP
ejde-761	542	15	the	the	DET
ejde-761	542	16	supports	support	NOUN
ejde-761	542	17	of	of	ADP
ejde-761	542	18	energy	energy	NOUN
ejde-761	542	19	solutions	solution	NOUN
ejde-761	542	20	of	of	ADP
ejde-761	542	21	quasilinear	quasilinear	PROPN
ejde-761	542	22	parabolic	parabolic	PROPN
ejde-761	542	23	equations	equation	NOUN
ejde-761	542	24	at	at	ADP
ejde-761	542	25	arbitrary	arbitrary	ADJ
ejde-761	542	26	order	order	NOUN
ejde-761	542	27	.	.	PUNCT
ejde-761	543	1	sb	sb	PROPN
ejde-761	543	2	.	.	PROPN
ejde-761	543	3	math	math	PROPN
ejde-761	543	4	.	.	PUNCT
ejde-761	544	1	,	,	PUNCT
ejde-761	544	2	190	190	NUM
ejde-761	544	3	(	(	PUNCT
ejde-761	544	4	1999	1999	NUM
ejde-761	544	5	)	)	PUNCT
ejde-761	544	6	,	,	PUNCT
ejde-761	544	7	18431869	18431869	NUM
ejde-761	544	8	.	.	PUNCT
ejde-761	545	1	[	[	X
ejde-761	545	2	43	43	NUM
ejde-761	545	3	]	]	X
ejde-761	545	4	strauss	strauss	PROPN
ejde-761	545	5	,	,	PUNCT
ejde-761	545	6	w.	w.	PROPN
ejde-761	545	7	;	;	PUNCT
ejde-761	545	8	wang	wang	PROPN
ejde-761	545	9	,	,	PUNCT
ejde-761	545	10	g.	g.	PROPN
ejde-761	545	11	;	;	PUNCT
ejde-761	545	12	instabilities	instability	NOUN
ejde-761	545	13	of	of	ADP
ejde-761	545	14	travelling	travel	VERB
ejde-761	545	15	waves	wave	NOUN
ejde-761	545	16	of	of	ADP
ejde-761	545	17	the	the	DET
ejde-761	545	18	kuramoto	kuramoto	NOUN
ejde-761	545	19	-	-	PUNCT
ejde-761	545	20	sivashinsky	sivashinsky	NOUN
ejde-761	545	21	equation	equation	NOUN
ejde-761	545	22	.	.	PUNCT
ejde-761	546	1	chin	chin	PROPN
ejde-761	546	2	ann	ann	PROPN
ejde-761	546	3	math	math	PROPN
ejde-761	546	4	b.	b.	PROPN
ejde-761	546	5	23	23	NUM
ejde-761	546	6	(	(	PUNCT
ejde-761	546	7	2002	2002	NUM
ejde-761	546	8	)	)	PUNCT
ejde-761	546	9	,	,	PUNCT
ejde-761	546	10	267	267	NUM
ejde-761	546	11	-	-	SYM
ejde-761	546	12	276	276	NUM
ejde-761	546	13	.	.	PUNCT
ejde-761	547	1	[	[	X
ejde-761	547	2	44	44	NUM
ejde-761	547	3	]	]	SYM
ejde-761	547	4	tao	tao	PROPN
ejde-761	547	5	,	,	PUNCT
ejde-761	547	6	y.	y.	PROPN
ejde-761	547	7	;	;	PUNCT
ejde-761	547	8	winkler	winkler	NOUN
ejde-761	547	9	,	,	PUNCT
ejde-761	547	10	m.	m.	NOUN
ejde-761	547	11	;	;	PUNCT
ejde-761	547	12	effects	effect	NOUN
ejde-761	547	13	of	of	ADP
ejde-761	547	14	signal	signal	NOUN
ejde-761	547	15	-	-	PUNCT
ejde-761	547	16	dependent	dependent	ADJ
ejde-761	547	17	motilities	motility	NOUN
ejde-761	547	18	in	in	ADP
ejde-761	547	19	a	a	DET
ejde-761	547	20	keller	keller	PROPN
ejde-761	547	21	–	–	PUNCT
ejde-761	547	22	segel	segel	NOUN
ejde-761	547	23	-	-	PUNCT
ejde-761	547	24	type	type	NOUN
ejde-761	547	25	reactiondiffusion	reactiondiffusion	NOUN
ejde-761	547	26	system	system	NOUN
ejde-761	547	27	.	.	PUNCT
ejde-761	548	1	math	math	NOUN
ejde-761	548	2	.	.	PUNCT
ejde-761	549	1	models	model	NOUN
ejde-761	549	2	methods	method	NOUN
ejde-761	549	3	appl	appl	PROPN
ejde-761	549	4	.	.	PUNCT
ejde-761	550	1	sci	sci	PROPN
ejde-761	550	2	.	.	PROPN
ejde-761	550	3	,	,	PUNCT
ejde-761	550	4	27	27	NUM
ejde-761	550	5	(	(	PUNCT
ejde-761	550	6	2017	2017	NUM
ejde-761	550	7	)	)	PUNCT
ejde-761	550	8	,	,	PUNCT
ejde-761	551	1	pp	pp	ADP
ejde-761	551	2	.	.	PUNCT
ejde-761	552	1	16	16	NUM
ejde-761	552	2	-	-	SYM
ejde-761	552	3	45	45	NUM
ejde-761	553	1	[	[	X
ejde-761	553	2	45	45	NUM
ejde-761	553	3	]	]	X
ejde-761	553	4	yoon	yoon	PROPN
ejde-761	553	5	,	,	PUNCT
ejde-761	553	6	c.	c.	PROPN
ejde-761	553	7	;	;	PUNCT
ejde-761	553	8	kim	kim	PROPN
ejde-761	553	9	,	,	PUNCT
ejde-761	553	10	y.	y.	PROPN
ejde-761	553	11	j.	j.	PROPN
ejde-761	553	12	;	;	PUNCT
ejde-761	553	13	global	global	ADJ
ejde-761	553	14	existence	existence	NOUN
ejde-761	553	15	and	and	CCONJ
ejde-761	553	16	aggregation	aggregation	NOUN
ejde-761	553	17	in	in	ADP
ejde-761	553	18	a	a	DET
ejde-761	553	19	keller	keller	PROPN
ejde-761	553	20	–	–	PUNCT
ejde-761	553	21	segel	segel	NOUN
ejde-761	553	22	model	model	NOUN
ejde-761	553	23	with	with	ADP
ejde-761	553	24	fokkerplanck	fokkerplanck	NOUN
ejde-761	553	25	diffusion	diffusion	NOUN
ejde-761	553	26	.	.	PUNCT
ejde-761	554	1	acta	acta	PROPN
ejde-761	554	2	appl	appl	PROPN
ejde-761	554	3	.	.	PROPN
ejde-761	554	4	math	math	PROPN
ejde-761	554	5	.	.	PUNCT
ejde-761	555	1	,	,	PUNCT
ejde-761	555	2	149	149	NUM
ejde-761	555	3	(	(	PUNCT
ejde-761	555	4	2016	2016	NUM
ejde-761	555	5	)	)	PUNCT
ejde-761	555	6	,	,	PUNCT
ejde-761	555	7	pp	pp	ADP
ejde-761	555	8	.	.	PUNCT
ejde-761	556	1	101	101	NUM
ejde-761	556	2	.	.	PUNCT
ejde-761	557	1	josé	josé	PROPN
ejde-761	557	2	luis	luis	PROPN
ejde-761	557	3	d́ıaz	d́ıaz	PROPN
ejde-761	557	4	palencia	palencia	PROPN
ejde-761	557	5	department	department	NOUN
ejde-761	557	6	of	of	ADP
ejde-761	557	7	mathematics	mathematic	NOUN
ejde-761	557	8	and	and	CCONJ
ejde-761	557	9	education	education	NOUN
ejde-761	557	10	,	,	PUNCT
ejde-761	557	11	universidad	universidad	PROPN
ejde-761	557	12	a	a	DET
ejde-761	557	13	distancia	distancia	PROPN
ejde-761	557	14	de	de	X
ejde-761	557	15	madrid	madrid	PROPN
ejde-761	557	16	,	,	PUNCT
ejde-761	557	17	28400	28400	NUM
ejde-761	557	18	madrid	madrid	PROPN
ejde-761	557	19	,	,	PUNCT
ejde-761	557	20	spain	spain	PROPN
ejde-761	557	21	email	email	NOUN
ejde-761	557	22	address	address	NOUN
ejde-761	557	23	:	:	PUNCT
ejde-761	558	1	joseluis.diaz.p@udima.es	joseluis.diaz.p@udima.es	PROPN
ejde-761	558	2	1	1	NUM
ejde-761	558	3	.	.	PUNCT
ejde-761	558	4	problem	problem	NOUN
ejde-761	558	5	description	description	NOUN
ejde-761	558	6	and	and	CCONJ
ejde-761	558	7	objectives	objective	NOUN
ejde-761	558	8	2	2	NUM
ejde-761	558	9	.	.	PUNCT
ejde-761	558	10	preliminaries	preliminary	NOUN
ejde-761	558	11	3	3	NUM
ejde-761	558	12	.	.	PUNCT
ejde-761	558	13	boundedness	boundedness	NOUN
ejde-761	558	14	of	of	ADP
ejde-761	558	15	solutions	solution	NOUN
ejde-761	558	16	4	4	NUM
ejde-761	558	17	.	.	PUNCT
ejde-761	559	1	travelling	travel	VERB
ejde-761	559	2	waves	wave	NOUN
ejde-761	559	3	4.1	4.1	NUM
ejde-761	559	4	.	.	PUNCT
ejde-761	560	1	travelling	travel	VERB
ejde-761	560	2	wave	wave	NOUN
ejde-761	560	3	instabilities	instability	NOUN
ejde-761	560	4	4.2	4.2	NUM
ejde-761	560	5	.	.	PUNCT
ejde-761	561	1	exact	exact	ADJ
ejde-761	561	2	travelling	travel	VERB
ejde-761	561	3	wave	wave	NOUN
ejde-761	561	4	profiles	profile	NOUN
ejde-761	561	5	and	and	CCONJ
ejde-761	561	6	characteristic	characteristic	ADJ
ejde-761	561	7	propagation	propagation	NOUN
ejde-761	561	8	speed	speed	NOUN
ejde-761	561	9	5	5	NUM
ejde-761	561	10	.	.	PUNCT
ejde-761	561	11	scaling	scale	VERB
ejde-761	561	12	invariance	invariance	NOUN
ejde-761	561	13	and	and	CCONJ
ejde-761	561	14	symmetry	symmetry	NOUN
ejde-761	561	15	6	6	NUM
ejde-761	561	16	.	.	PUNCT
ejde-761	562	1	conclusions	conclusion	NOUN
ejde-761	562	2	references	reference	NOUN